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To
MARGOT

" But that which relies on calculation and
measurement will be the best element in the soul?"

" Of course."

11 Then that which opposes it will be one of the
beggarly elements in us?

" Inevitably."

PLATO, Republic.

Christian Huygens

CHRISTIAN HUYGENS

AND

THE DEVELOPMENT OF SCIENCE
IN THE SEVENTEENTH CENTURY

By

A. E. BELL. Ph.D., M.Sc.

Head of the Science Department, Sandhurst
Formerly Head of the Science Department, Clifton College,

LONDON

EDWARD ARNOLD & CO.

COPYRIGHT

First published 1947
Reprinted 1950

Printed in Great Britain by

Sons Ltd., Guild ford and Es/ier

PREFACE

THERE can be no doubt that Christian Huygens was one of the
greatest scientific geniuses of all time. A man who transformed
the telescope from being a toy into a powerful instrument of in-
vestigation, and this as a consequence of profound optical
researches; who discovered Saturn's ring and the satellite Titan;
who drew attention to the Nebula in Orion; who studied the prob-
lem of gravity in a quantitative manner, arriving at correct ideas
about the effects of centrifugal force and the shape of the earth;
who, in the great work Horologium Oscillatorium, founded the
dynamics of systems and cleared up the whole subject of the
compound pendulum and the tautochrone; who solved the out-
standing problems concerned with collision of elastic bodies and
out of much intractable work developed the general notion of
energy and work; who is rightly regarded as the founder of the
wave theory in light, and thus of physical optics such a man
deserves memory with the names of Galileo and Newton, and
only the accidents of history have so far prevented this. It might
be argued that Robert Hooke, who like Huygens was influenced
by Descartes 's teachings, is of equal stature and showed as much
inventive genius and intuition. In the extraordinary range of
their activities there is some similarity. The overwhelming
difference lies in the fact that Huygens was a great mathema-
tician and exponent of the quantitative method, whereas Hooke
could never get beyond the first phase of a piece of work : that
which led to the need of exact measurement and the discovery of
mathematical relations.

Having made this claim for Huygens, it is natural to ask how
he compares with Newton. It is a question which arises from
time to time in these pages, and one to which there is no
epigrammatic answer. In some senses it was Huygens's greatest
misfortune to grow up under the powerful influence of Descartes,
who was a grfeat friend of his father, a frequent visitor to the

5

6 THE LIFE OF CHRISTIAN HUYCENS

family, and master of at least one disciple who taught Huygens
at the university. From Descartes too many of Huygens's
hypotheses were taken, so much so that he might stand as the
exact opposite of Newton, whose objection hypotheses non fingo
he did so much to call forth. Looking at Huygens in this way it
is easy to dismiss him as a " Cartesian ", one whose ideas were
largely superseded after the publication of Newton's Principia.
But this would be a serious mistake. If he could not so
brilliantly see the way to extend the sphere of natural law to
the heavens, perceiving that the task of science is not to
disclose a mechanism so much as to arrive at laws, he neverthe-
less did important work to prepare men of science for this
modern attitude. If Newton owed nothing to Huygens, and he
certainly owed exceedingly little, it is very probable that he was
indebted in another way, for it may well have been the feeling of
dissatisfaction with the position men like Huygens were reaching
that drove Newton to make the new " instauration " Bacon had
looked for a renovation of natural philosophy. The progress
of scientific explanation may then be seen to be a process of
leaving out redundant elements, of emancipation from imaginary
qualities, until one arrives at the really successful procedure of
abstraction.

But Huygens was in all other senses an astoriishingly modern
thinker, and he had the disposition which sets out to face things
as they are which marks the man of science as much as does the
possession of specialized knowledge. As a scientific researcher
he was the first of a new profession, and this permanent occupa-
tion with science brought its own characteristic attitude of mind.
Colbert, Louis XIV's energetic and shrewd minister, subsidized
scientific investigation for the first time in history when he
obtained pensions for Huygens and the other scientists who
formed the nucleus of the Academic Royale des Sciences. Of
course Colbert had his eyes on commercial as much as on
intellectual advances. Considering his jealousy of Dutch com-
merce Colbert was indeed fortunate to get as an ally of French
power the most noted Dutch scientist of the age. It has indeed
been a puzzling feature of Huygens's life that, having embraced
French manners, delighting in the freedom that his position at
the Biblioth&jue du Roi gave him, he could even so remain in
Paris in his official position during the years when Louis waged
war on the Netherlands, determined if he could to break the

PREFACE 7

newly found independence of the Dutch for ever. More than
this, Huygens in 1673 dedicated his magnum opus, the Horolo-
gium Oscillatorium, to his royal patron.

Huygens possessed a delicate constitution and was, it should
be admitted, not of the stuff of which fighters are made. Like
Pascal he suffered from frequent illness, like Spinoza there was a
certain effeminacy about him. Again it may be argued that war
in those days never concerned more than the limited class of
professional soldier. If Huygens had quitted Paris the best he
could have done towards the defence of Holland would have
been to work in a diplomatic capacity as did his brother
Constantin, or decipher codes as did the mathematician John
Wallis in the civil war in England, or be killed like any ordinary
soldier as was Gascoigne at Marston Moor. Men of the profes-
sional class in those days were not expected to give up their
activities, and there can be little doubt that Huygens's scientific
work benefited greatly from his life in Paris. At home at
Voorburg, near the Hague, he lived the life of a recluse with only
this stimulus of his correspondence with Paris and London. The
investigation of the physical world appeared to Huygens and to
nearly all men of science to be something of such moment that all
events in the social and political world were merely so many
inconvenient interruptions. That this was so in the seventeenth
century is evident from the way in which the meetings which led
to the foundation of the Royal Society were carried on during the
civil war. From the very beginning the men of science felt them-
selves to be advancing the real causes of humanity and this
longer view may well have been the one at which Huygens looked
when, in 1672, he decided to remain in Paris.

From birth Christian Huygens grew up under the strongest
French influences. In Paris, as a young man, he evidently
imbibed the fashionable ideas which in religion tended to
agnosticism and in morals pointed the way to greater freedom.
His seriousness forbade the lax immorality then current. " The
beaux esprlts believed in God merely as a matter of courtesy and
for reasons of state ", wrote Garasse, but while this might be
true, the men of science, especially in England, kept their
religion. Huygens stands in contrast with the other great scien-
tists of his time and in closer resemblance with some of the
encyclopaedists of the following century in that he turned away
from Calvinism as from Catholicism.

8 THE LIFE OF CHRISTIAN HUYGENS

In spite of the evident growth of power politics and the
existence of widespread corruption, the period of Huygens's life
was one of extraordinary optimism among the men of science.
Science was the last activity to come from the humanistic
impulse and it was to its devotees the most dazzling discovery to
be attributed to man's freedom. "... this is the age ", wrote
Henry Power, " wherein all men's souls are in a kind of fermen-
tation, and the spirit of wisdom and learning begins to mount
and free itself from those drossie and terrene Impediments
where-with it hath been so long clogg'd . . . This is the age
wherein (methinks) philosophy comes in with a Spring-tide ..."
The greater minds of the period were less confident over the in-
evitability of progress; Huygens in particular was especially
cautious when asked to compare his own times with the age of
Pericles. But a cruder spirit came to be associated with the men
of science of the eighteenth and nineteenth centuries.

At least one modern writer has, however, condemned the
" false modern emphasis " on the bold confidence and rebellious
energy of the Renaissance, and has pointed out the amount of
pessimism in English seventeenth century literature. On the
Continent the inroads on religious belief seem in fact to have
been more rapid than they were in England. Bishop Sprat,
writing a defence of the Royal Society in 1667, believed that
science would never undermine the socially acceptable beliefs of
his time. He never dreamed that a "Universal Inquiry into things
hitherto unquestioned " could have the unending consequences
it has in fact produced. He obviously supposed, as did Descartes,
though not Huygens, that the complete scientific account of the
universe would in its essentials require the labours of only one or
two generations of gifted intelligences. In these labours the
experimentalists would, thought Sprat, have always before their
eyes the " Beauty, Contrivance and Order of God's works ". As
the Master of Trinity College has written, " God was to be
praised by studying the plan of His creation, but no further
attempt was to be made to fit the findings of science into the
scheme of theology, as the schoolmen of old had striven so long
and painfully to do." This was Newton's position, it was
Boyle's, but it was not Huygens's. He alone among the men of
science of his day found the temper of scientific enquiry alien to
orthodox religious faith. As it was, both in France and England
I here were divines who supported the plan of organized enquiry

PREFACE 9

in natural philosophy. Liberal minded abbes and protestant
bishops gave their approval to scientific research for the glory of
God and the service of man. Occasionally the former tended to
lose their nerve and clung then for stability to Descartes's
" system ".

The particular feature of Huygens's thought which was at
the same time his strength and weakness was his concern for
particular problems and his distrust of all speculative general-
izations. This distrust he no doubt acquired during his study of
Descartes's writings. His attitude towards Spinoza's ideas may be
explained in this way, for it would otherwise have been expected
that these ideas would have gained his sympathy, for Spinoza,
of all great religious writers, has come nearest to expressing the
scientific attitude to the world. But Huygens distrusted the
system which Spinoza elaborated and no doubt for the reason
that Spinoza sought to apply the Cartesian method. This
method, Huygens saw, is unable to give us an understanding of
nature, and he himself made great contributions to the new
scientific method. The fact that he personally despised Spinoza
seems to be explained by a sense of superiority which rested on
social rather than on intellectual considerations. It was not, of
course, a democratic age.

In his scientific work Huygens was the greatest mechanist of
the seventeenth century. He combined Galileo's mathematical
treatment of phenomena with Descartes's vision of the ultimate
design of nature. Beginning as an ardent Cartesian who sought
to correct the more glaring errors of the system, he ended as one
of its sharpest critics. The development in the seventeenth
century of Dynamics and Astronomy beyond the stage of
geometrical description demanded new inductive principles of
correlation; the ideas of mass, weight, momentum, force, and
work were finally clarified in Huygens's treatment of the phen-
omena of impact, centripetal force and the first dynamical
system ever studied the compound pendulum. In Astronomy
Huygens explained the appearances of Saturn, until 1656 the
greatest anomaly of the Copernican system. His eminence as an
observer was due to the high quality of his telescopes and this, in
part, resulted from his thorough theoretical researches on the
problems first attacked by Kepler, Snell and Descartes. It is well
known that physical optics practically took its rise from
Huvgens's Traite de la Lumi&re.

IO THE LIFE OF CHRISTIAN HUYCENS

The growth of interest in the history of science may be con-
sidered to be in a sense an outcome of the increasing specializa-
tion of science itself. So much is commonly ignored in concen-
trating on the discipline of science, that for education in the
broader sense, when once the demands of life are allowed to
supersede merely professional ones, something more is needed.
Many have felt that the history of science may provide an im-
portant humanistic element. A protest may here be made against
the practice of inserting historical notes in scientific textbooks
without regard for the conflict of old and modern ideas. Once a
subject has become well developed, the logical and not the histor-
ical method is to be desired, for so much of the earlier science can
be properly understood only with a really adequate study. The
great scientist of the past tends soon to appear a distant and
indeed thoroughly dead sort of figure. The modern scientist to
a large extent accepts his reputation on trust and has little time
and often less inclination to read the original work. How many
now read Galileo's Discourses or Newton's Principiaf It is other-
wise in "art and in literature. If science is to become a more
widely accepted means of education (in the sense of a form of
culture) there is need of more works on its history. A modern
estimate can do much to revivify the past and present these great
men, its creators, in a clearer light.

As an account in English of the work of Huygens this study
is to be regarded as only a beginning. The Oeuvres Completes de
Christiaan Huygens, published by the Societe Hollandaise des
Sciences, comprise more than twenty magnificent volumes and
contain all the material for a definitive work; beside them the
present book must appear almost insignificant. It is greatly to be
hoped that before long a large work on this great subject will be
written by a scholar of the requisite stature. Writing as one whose
training has been principally in chemistry the author has met with
many difficulties in Huygens's works. It need hardly be said that
they are properly to be studied by a mathematician, while the
subject as a whole requires a combination of historian and Latin
scholar, physicist and philosopher, which it must be rare to
achieve.

A.E.B.

CLIFTON, 1947

CONTENTS

Page

PREFACE 5

PART I

NOTES ON THE LIFE OF CHRISTIAN HUYGENS . . . 13

PART II

I. THE STATE OF SCIENCE IN THE FIRST HALF OF THE

SEVENTEENTH CENTURY 97

II. WORK ON COLLISION BETWEEN ELASTIC BODIES . 109

III. CENTRIFUGAL FORCE 117

IV. STATICS 124

THE TREATISE ON THE PENDULUM CLOCK :

The Horologium Oscillatorium
V. Part I. Construction and Use of the

Pendulum Clock . . . . 127

VI. Part II. Oscillation in a Cycloidal Arc . . 136

VII. Part III. Evolutes and the Measurement of

Curves 145

VIII. Part IV. The Centre of Oscillation of a Com-
pound Pendulum . . . . 150

IX. THE CAUSE OF GRAVITY 161

X. HUYGENS'S OPTICAL STUDIES 165

XI. THE WAVE THEORY OF LIGHT . . . . 176

XII. SATURN 193

XIII. COSMOTHEOROS 2OO

XIV. THE PLACE OF HUYGENS IN THE HISTORY OF SCIENCE 203

PERSONS MENTIONED 213

BIBLIOGRAPHY 217

INDEX 219

ii

PLATES

Christian Huygens frontispiece

facing page
I. Saturn reproduced from Huygens's MS 32

II. One of Coster's clocks 38

III. Huygens's Clock as the centre feature of a design

showing scientific apparatus of 1671 56

IV. Members of the Academic Royalc dcs Sciences 58
V. Louis XIV at a meeting of the Academic 60

VI. Drawing by Huygens of his vacuum pump, 1668 162

PART I
THE LIFE OF CHRISTIAN HUYGENS

CHRISTIAN HUYGENS has been a strangely neglected figure
apart from the study he has rightly received in his native
Holland. A man of the greatest scientific genius without any
doubt, he was one in whom great sagacity and mathematical
power went side by side with a feeling for elegance and form in
the interpretation we make of Nature, so much so, that it is with-
out surprise that we find he was devoted to music and the arts
ajid was by no means the type of narrow research worker that
later scientific studies did for a time produce, and still produce
in some measure. Huygens was a professional scientist in an age
when the boundaries of Science were scarcely drawn, and his
interest lies as much in his general outlook as in his specialized
studies.

Huygens had not the religious feeling of a Spinoza or the
sensitivity of a Pascal, nor was he a philosopher of the stature of
Descartes or a mathematician of the rank of Leibnitz. In an age
when the human mind was making great marches into the
territory of natural philosophy, Huygens's energies were thrown
now into the study of applied mathematics, now into optical
researches or astronomy; and he managed somehow to pursue
the most strikingly original researches in several subjects quite
simultaneously, so that in his note-books matters of the most
varied kind jostle one another in profusion, and a very large
volume indeed would be needed to do justice to his labours.
What is of chief significance to-day can be reduced to much
smaller limits, and the reader who wishes for more must go to
the great volumes published by the Societe Hollandaise des
Sciences under the auspices of the Dutch Government.

Here we are concerned rather to look back for a space on that
interesting period in Europe between the death of Galileo in

'3

14 THE LIFE OF CHRISTIAN HUYGENS

1642 and the rise to fame of Newton, a period in which
Huygens, in fact, stood unchallenged as the greatest man of
science of the age.

It has been remarked 1 that " In 1600 the educated English-
man's mind and world were more than half medieval; by 1660
they were more than half modern ". And this remark need not
have been limited to Englishmen. On the Continent also, about
the middle of the century, a certain profoundly important
change was becoming visible. It was, perhaps, in the years
following 1670 that the break-away from authoritative teachings
of Descartes as of the schoolmen became the feature of the
really important scientific theories. Galileo and Huygens both
struggled to make use of teaching they received in their youth,
and both failed; they were each forced in some degree to rely on
their own powers. Indeed, underneath all successful scientific
work there lies a great deal of experiment in failure.

One must read Dante, or toil over Thomas Aquinas, to get
a picture of the universe as it was conceived by educated men in
the Middle Ages. The sheer verbalism of all argument about the
world repels and astonishes the modern reader, but there was an
undeniable attractiveness in the notion of a Cosmos : the " idea
of a hierarchically-ordered finite world structure ", a world in
which all was made for man and consequently one in which
clear and simple reasons existed why things are as they
are. What we see as an appeal to objectivity must then have
seemed to some to be pure obstinacy and blindness, for what the
men of science really abolished was not so much an over-rational
world structure as the appeal to feeling in the making of explana-
tions. The new studies offered at first no more satisfaction than
that which could be found in the agreement of theory with
measurement. Nevertheless, scientific explanations did not get a
reputation for their " inhuman " quality until the eighteenth
century, when many physical theorems were generalized in
abstract mathematical form.

Early in the century Descartes worked out an ingenious and
even aesthetically satisfying system which welded natural science
on to the structure of a philosophical theory about the nature of
matter and of space, and some reference to this system must be
made in later pages. The chief point about Descartes's teaching,

Douglas Bush : English Literature in the Earlier Seventeenth Century,

THE LIFE OF CHRISTIAN HUYGENS 15

if it were accepted, was that experiment and observation could
soon be dispensed with and the human mind could rest satisfied
with the knowledge it could gain through a rationale worked out
by philosophers. So seductive was his reasoning, and so per-
suasive the arrangement of the arguments, that both in France
and England there were soon many ardent Cartesians who were
distinguished by the ease with which they accounted (in a
general way) for natural phenomena. Since space was supposed
to be full of a " subtle matter " and this moved around each
planet in a kind of vortex, it was easy to imagine various
effects as resulting from the properties of this medium. And
Huygens was himself for many years a Cartesian. The essays
produced by Descartes were a flirtation with the mathematical
treatment of observations begun by Galileo, only they went far
further and cast the human mind in great voyages of imagina-
tion further, in fact, than it was yet ready to go. It is always an
interesting question, therefore, how Huygens came to be a strong
critic of Cartesianism, and on the other hand, why he rejected
Newton's treatment of gravitation and even at the end of his life
had not thrown overboard the whole Cartesian apparatus. Of all
the events in Huygens's life when one would give much to know
what happened, there is an occasion of which one gets only a faint
glimpse: Huygens and Newton getting into a stage coach at
seven o'clock on a July morning in 1689, to go from Cambridge
to London, Huygens was then sixty and his zeal and lively
curiosity were unabated; Newton was forty-seven, and every-
where acclaimed as the author of the magnificent Principia
though it had to be confessed that only a handful of men really
knew what it was all about. Huygens had left Holland in poor
health in order to see Newton and to visit old friends among the
English men of science. But all that can safely be connected with
this meeting is the fact that Newton subsequently produced a
further study of the Cartesian vortices and, on the other hand,
Huygens began to object to Leibnitz's use of them. As the coach
rolled on its way to London, it may be that Huygens was turn-
ing over in his mind the final objections to any further develop-
ments of Descartes's ideas. His own work had led far in this
direction and the end of it all seemed to be that Descartes's
ventures in physics had been pure romance, " un beau roman de
physique " as Leibnitz himself concluded. Or is such conjecture
too dangerous? Huygens, with his only moderate English and

16 THE LIFE OF CHRISTIAN HUYGENS

his weakness for a picturable sort of explanation may have made
little of a taciturn Newton; he recorded nothing of interest from
the meeting.

II

Unlike Newton, Christian Huygens came of a family which
had already shown genius. His father, Constantin, was extra-
ordinarily brilliant; a poet, student of natural philosophy,
classical scholar and diplomat, he typified the conception of
culture at its best at the beginning of the century. As secretary
to the Prince of Orange, Frederick Henry, he must be considered
important in the guiding of the country through difficult times.
In this, however, his own father, an earlier Christian, stood as an
example, for he had been secretary to William the Silent in the
eventful years after 1578. This Christian was a native of the
Southern Low Countries, while his wife, Susanna Hoefnagel, was
of Antwerp, though at the time of her marriage a protestant
refugee from Amsterdam. The two sons, Maurice and
Constantin, were born in troubled times, the latter on September
4th, 1596, at the Hague.

The last quarter of the sixteenth century saw the indepen-
dance of the seven northern provinces of the Netherlands
regained after an eighty years' struggle with Spanish power. In the
South, Spain and Catholicism continued to dominate; in the
North, religious and political liberation occurred together and
there grew up a deep mistrust of all hierarchial powers; even the
doctrines of Luther were rejected because they acknowledged the
authority of the State in religion. A new Calvinist common-
wealth now existed, and its rise has been described in the pages
of Motley's Rise of the Dutch Republic. The assassination of
William the Silent, in 1584, came after he had accomplished his
great task for he had, as Motley says, " planted a free common-
wealth under the very battery of the Inquisition in defiance of
the most powerful empire existing ".

It is interesting to look back at these important events which
came close to the life of the Huygens family. When Maurice of
Nassau was engaged in defeating the Spaniards in the open field,
Constantin Huygens, father of the scientist, was receiving a care-
ful and thorough education as a boy. This Huygens showed
quite a distinction in mathematical work but all the influences

THE LIFE OF CHRISTIAN HUYGENS 17

of his life were in the direction of the courtier and diplomat. He
was often at the court of Louise de Coligny, the widow of
William the Silent, and he accordingly spoke French from boy-
hood. He completed a course of Law at Leyden University and
then was introduced at twenty-one to the life of diplomacy. This
Huygens became by far the most well-known member of the
family up to the mid-seventeenth century. His all-round culture
has been mentioned, and he did in fact become known all over
Holland, and in England, as a latinist and poet, as an amateur
of music and painting, and as a student of philosophy. He was,
besides, a close friend of Descartes and, at length, best known
of the leaders of contemporary thought in the Republic : " no
Dutchman commanded a more European culture; no Dutchman
was more thoroughly Dutch ". After their first meeting,
Descartes wrote of him " . . . despite what I heard of him, I
could not believe that a single mind could occupy itself with so
many things and acquit itself so well of them all ".

Christian Huygens's father, then, was a man of outstanding
ability and brilliance and he was very well known in England.
He studied at Oxford for a time and became an intimate friend
of John Donne. He played the lute at the court of James I, and
in 1622 received an English knighthood. Nor was this brilliance
a mere glitter, the effect produced by a versatile and fashionable
courtier. Constantin Huygens corresponded for years with
Descartes, with Mersenne, the great intermediary of men of
science of that time, with Diodati, a friend of Galileo, and with
many well-known mathematicians, notably Schooten the elder.
In his MSS. have been found notes on Euclid's propositions and
records of his study of optics. When Golius succeeded Snell at
Leyden, Constantin Huygens recommended him to apply him-
self to optics. "The consequences of the law of refraction
[formulated by Snell in 1621] have not been sufficiently studied
by anyone," lie wrote. He himself is said to have attempted to
grind lenses to the forms proposed by Descartes the surfaces
being of elliptical or hyperbolic section instead of spherical.
Descartes had concluded that such lenses would be free from
spherical aberration but Huygens (or the skilled mechanic
employed by him) found the work impossible with the ordinary
tools then used. His indirect influence in scientific work was
undoubtedly of greater significance : without his encouragement
Descartes might never have published his Dioptrique. The

18 THE LIFE OF CHRISTIAN HUYGENS

philosopher was induced to overcome his well-known
hesitancy only through the efforts of Constantin Huygens and
Mersenne.

This versatile man of letters and diplomat in 1627 married his
cousin, Susanna van Baerle, daughter of a wealthy merchant of
Amsterdam and by all accounts an intelligent and cultivated
woman. The children of this marriage, which must interest
geneticists, were Constantin (1628), Christian (1629), Louis and
Philip, the last of whom died young, and Susanna. In 1637, after
only ten years of married life, the mother of this family herself
died. Another cousin took over the care of the family, which
removed to a newly built country house at Voorburg, close to the
Hague. Here, when he had recovered from the death of his wife,
Constantin received officers of the French army, French
diplomats and men of letters. Here Descartes himself made
occasional visits and remarked on the prowess of young Christian
in mathematics, a study in which he complained he saw no great
progress.

Descartes spent a good deal of time in Holland and did much
of his more important work in the quiet of the country. Even in
Holland, however, he did not feel sufficiently secure to bring out
his treatise Le Monde and it was not until 1637 that his Discourse
on Method appeared. But it is easy to imagine the great influence
of Descartes on the intellectual family at Voorburg in those
years just succeeding the publication of the famous Discourse.
The work itself shows the appeal of Descartes's mode of argu-
ment and, to a generation who read and sympathized with
Campanula's Defence of Galileo, it must have seemed that
Descartes was indeed the apostle of intellectual freedom.
Campanella's tract, composed in a Neapolitan dungeon in 1616,
was printed at Frankfurt in 1622, and during the next thirty
years it was widely read by educated men all over Europe. Its
courageous stand for freedom of enquiry and for the truth of
the Copernican theory was a source of inspiration. For it is clear
that a generation which could revere Galileo did so because their
minds were alreadly partly prepared by earlier critics of Aristotle:
Benedetti, Stevinus and others. In Campanella there was a
vigour and boldness which recalled Giordano Bruno. Bruno and
Campanella held that there are an infinite number of worlds, and
if in Descartes's writings this doctrine as well as that of
Copernicus was taught with great caution there can be no doubt

THE LIFE OF CHRISTIAN HUYGENS 19

that this was through circumspection. Descartes was a cautious
man but very probably in conversation he was bolder.

Constantin Huygens was extremely proud of his two eldest
sons, who early showed intellectual brilliance. They were taught
at home by a private tutor until Christian was sixteen. This
education included singing, playing the lute, and the composi-
tion of Latin verses. Like Newton, as a young boy Christian
loved drawing and the making of mechanical models on which
he spent much labour and ingenuity. So much so that his tutor
felt misgivings; such practical work was after all an inferior and
even a dubious sort of occupation for a young man of family and
position. From the beginning, however, Christian showed
promise of great skill in geometry while his brother, Constantin,
excelled rather in literary compositions. Descartes was much
impressed with some very early work of Christian's and he saw
that great things might be expected from this rather serious boy
with the rather pale face and the large dark eyes. Christian was
rather delicate and by nature gentle, and his sensitivity seemed
almost feminine to his father, who seems to have been fortunate
in possessing an unflagging and exuberant vigour, quite different
in character from his son.

Characteristically enough, the first experiments of the youth-
ful Christian were in mathematics, and this is typical of him, for
he rarely ventured publications on other than abstract and some-
what theoretical subjects. But the influence of a cultured and
enlightened society remained with him, and his interests, early
determined, lasted unchanged all his life.

In 1645, when he was sixteen, Christian and his brother
entered the University of Leyden. Here they studied Mathe-
matics as well as Law, the younger Schooten, a protege of
Descartes's, then being professor. Schooten was an able
mathematician and Christian acquired the reputation of being
his best pupil. Mathematics was a subject which included what
we would now call mechanics and, for example, in Stevin's
Hypomnemata, a work in six volumes, there are discussions of
centres of gravity, levers, simple machines and hydrostatics.
Christian's father was clear about the supreme importance of
mathematical training. In 1644 Descartes had published his
Principia, a bold attempt to reduce all the changes of Nature to
mechanistic processes and he, it was well known, exalted the
study of the subject. Radical changes were taking place in men's

2O THE LIFE OF CHRISTIAN HUYGENS

ideas and during his time at Leyden Christian lived in an
atmosphere of intellectual ferment. The ideas of Descartes were
hotly contested by the Aristotelians and to such an extent that
in 1646 and 1647 the university almost became a battlefield. Un-
fortunately, there are only scanty records of Huygens's reactions
to these experiences. Regarding Descartes's Principia he many
years later remembered the deep impression it made on him. " It
seemed to me when I first read this book, the Principia, the
first time," he wrote, " that everything in the world became
clearer and I was sure that when I found some difficulty that it
was my fault that I did not understand this thought. I was then
only fifteen or sixteen years old/'

Descartes's ideas were strongly represented in Holland. Renier,
one of his disciples, taught Cartesian philosophy at Leyden for a
time and later went to Utrecht. Here 'he had great influence and
was followed by Regius, one of his own pupils. Aristotelian
philosophy was associated with the Jesuits and nowhere more
than in northern Holland was their influence more strongly re-
sisted. Nevertheless, even in Holland, freedom of thought was not
absolute and only a few years previously the Aristotelians had
scored notable victories by arousing suspicion as to the religious
consequences of Descartes's teachings. Cartcsianism owed its wide
appeal to the ned felt for a new celestial mechanics after the
acceptance of the ideas of Copernicus. This, apparently,
Descartes's theory supplied. Moreover, Aristotle's outlook in
natural science was in the main teleological. It was felt that if the
guiding principle of teleology were abandoned some way of
expressing the determinism of events must be found. On this
point Descartes's analysis proved less sound but his system as a
whole was ingenious and even aesthetically satisfying.

In 1647, after two years at Leyden, Christian Huygens joined
his brother at the College at Breda. This college, founded by
Frederick Henry, seems to 'have achieved a temporary fame but
it came to an end during the century. Descartes seems to have
taken some interest in the place and certainly the forces of Aris-
totelianism were there unable to challenge the new philosophy.
John Pell, an Englishman, taught mathematics and was a man
of quite high reputation. It was fortunate that, after Schooten,
Huygens had so able a teacher.

As soon as Huygens's period at Breda was completed he made
a number of journeys, first going to Denmark in the company of

THE LIFE OF CHRISTIAN HUYGENS 21

the Count of Nassau-Siegen and later, with Constantin, to Frisia,
Spa and Rome. When in Denmark it was a great disappointment
to him that the weather made it impossible to reach Stockholm
for Descartes was then living at the court of Queen Christina.

Travel and a thorough education were, however, not the only
elements which made up the pattern of Christian's early years.
Most important, perhaps, of all was the correspondence he took
up with Pere Mersenne, who was next in importance to Des-
cartes among his father's acquaintance in the centre of the
learned world. Duhem has described Mersenne as a man of in-
satiable curiosity and the exuberant imagination of the artist.
He was at this time the great intermediary for scientific com-
munications between the chief centres of experiment. He popu-
larized much of Galileo's work and did much to thrash out those
fundamental notions on which seventeenth century mechanics
was based. Men like Descartes, Gassendi, Fermat and Pascal met
together at the cell of the Minorite father in Paris and this group
has been described as the origin of the Academic Royale des
Sciences. Mersenne was indeed a remarkable man, for he retained
the esteem of both Church and the scientific world ; "... he
did not believe all his religion," Pineau wrote to Rivet, " he was
one of those who are glad enough to see church service done . . .
he dared not often repeat his breviary for fear of spoiling his good
Latin." He was not himself a great originator. Pascal possessed
for mathematical and scientific work all the qualities which Mer-
sennet lacked: a profound penetration, logical rigour, critical
acuity, but Mersenne saw clearly which problems then mattered
most and Huygens was indebted to him for many of the subjects
of his early researches.

Aristotle, whose mechanics was the weakest part of his natural
science, had supposed that heavy bodies fall towards the centre of
the earth because this is their " natural " place. The heavier a
body is, the faster it moves towards the earth. If it were to fall
through a hole passing through the centre of the earth it would
come to a standstill on reaching the centre. As early as 1585 Bene-
detti had protested against this. He saw, in a general way, that
the inertia of the mass would carry it past the midpoint and that
it would in fact oscillate after the manner of a pendulum bob.
Stevin, with greater certainty than in the case of Galileo, is
known to have experimented by dropping large and small
weights simultaneously and showing that they reached the

22 THE LIFE OF CHRISTIAN HUYGENS

ground together. Galileo made a more thorough examination
of naturally accelerated motion and calculated the distances tra-
velled in successive seconds by a freely falling body. Mersenne,
in an early letter to Huygens, questioned if in fact the mass did
not in some way determine the limit of the velocity which could
be imparted. Huygens explained that his objections were all
based on observations of air resistance and gave such an able ex-
position of what is now termed Newton's first law of motion that
Mersenne gave him ungrudging praise : " I assure you that I
think so highly of your demonstration concerning falling bodies
that I believe Galileo would have been delighted to have you as
his follower/' Mersenne went on to set Huygens the problem of
finding the form taken up by a rope hanging from its two ends
which are fixed at the same height and some distance apart. Huy-
gens did not solve this mathematical problem until he recurred
to it late in life but he studied the disposition of weights along
the rope which would give it a parabolic form. He also became
interested in Mersenne's famous problem of determining the
centre of percussion of suspended bodies. This most important
problem was given its first general solution by Huygens many
years later.

Young Huygens was delighted with these letters, which he
received " with joy and avidity ". His father noted with approval
the penetration with which young Christian, then only seven-
teen, tackled problems then exercising the world's foremost men
of science. In December 1646 Christian wrote that he was occu-
pied with problems of centres of gravity and with modern de-
monstrations of some of Archimedes' propositions on the sphere
and cylinder a remark which illuminates the nature of his early
training " but nothing yet concerning centres of percussion of
which you recently wrote. However, I shall not fail to do all that
I can to find the demonstration although, up to the present, it
seems to me to surpass my ability . . ." Mersenne acknowledged
that he also could not see how a single rule could satisfy the
variety of figures for which the centre of percussion (or of oscil-
lation) was required. The problem was that of finding a formula
which would make it possible to calculate for any suspended
body the length of the simple pendulum which would have the
same period of oscillation. An experimental solution could, of
course, be found but this was not acceptable as an answer. It
is at first sight surprising that a grert deal of interest should be

THE LIFE OF CHRISTIAN HUYGENS 23

aroused by so academic a problem. This was because the prob-
lem was one concerning a dynamical system (as opposed to a
single mass) and it was obvious that a new approach was needed.
Problems of this sort led to the development of the calculus by
Newton and Leibnitz. Huygens, however, obtained a solution
in advance of either of them, although it was without the use
of their modern methods.

Some interesting matters are discussed in the correspondence
of Mersenne and the elder Constantin Huygens. Christian,
because of his precociousness, is sometimes referred to as a mod-
ern " Archimedes ". Mersenne wrote about the new work of the
young Pascal, then twenty-five, the problem of the nature of the
vacuum, the development of the telescope and the most recent
astronomical observations. There was a widely accepted belief
that a true vacuum is contrary to nature and this made it very
necessary to explain the well-known experimental results
obtained by Viviani and Torricelli in 1643.

The followers of Descartes were in obvious difficulties because
Descartes rejected the atomic doctrine and with it the notion of
a void. Since Gassendi was reviving, at least in part, the atomic
doctrine of Epicurus and considered it a profound philosophical
necessity that a vacuum should be possible, this apparently recon-
dite and academic matter aroused vigorous controversy. Galileo
attributed the more or less constant height of the barometer to
an equilibrium between the weight of the column of liquid
and an attractive " force " acting upwards. This force was, of
course, quite an illusion. Torricelli and Viviani verified that the
relative heights of liquid which could be supported in a baro-
meter tube varied inversely with the densities and in 1644 Torri-
celli really gave the correct explanation based on the pressure of
the atmosphere. Four years later Pascal's explanation of the
behaviour of the barometer was put forward, after the experiment
carried out for him on the Puy-de-D6me in September 1648, but
his views were by no means universally accepted. Quite a litera-
ture was produced in disproof of the existence of the vacuum
and even Constantin Huygens, who confessed himself most anxi-
ous to " penetrate all the mystery ", found the new explanation
very contrary to his inclinations, which were all for Cartesianism.

It is important to realize the fascination which all but a few
critical spirits found in Descartes's natural philosophy (see p. 109).
Tn a sense, of course, Cartesianism was anti-scientific. At a time

24 THE LIFE OF CHRISTIAN HUYGENS

when the trend of natural philosophy was in the direction of
empiricism, Descartes emphasized the great limitations of the
empirical method. While he scorned scholastic logic he con-
sidered that mere empiricism was futile and that his discovery of
analytical geometry illustrated the true method by which physi-
cal problems of all kinds involving motion in space could be
attacked. He believed that the way was open to reduce all phe-
nomena to the terms of geometrical description. From the pos-
tulates of space and motion, without any assumptions as to the
innate properties of matter, he hoped, by successive applications
of his intuitive method to isolated problems, to build up an
account of all the phenomena of the Cosmos. The a priorism of
Descartes's method is thus anti-scientific. On the other hand it
must be remembered that Aristotelian science was concerned with
logical rather than spatial relations. Descartes, on the other hand,
has been well described by the remark that he was " the author
and prophet of mechanism". With Galileo he asserted the
belief that the laws of Nature are both simple and open to dis-
covery. The danger for a youthful student such as Huygens was
that Descartes paid too little regard for what have been called
" stubborn and irreducible facts " and that he strayed too far from
the path of scientific work in undertaking to heal the schism
between the natural and the revealed.

In the famous vortex theory of the Principia Philosophize
(1644) Descartes supposed all space to be filled with a " subtle
matter " which moved with the planets in their paths. He made
brilliant play with this medium and used it to work out plausible
explanations of gravity and magnetism as well as the action of
the barometer. Light was treated as an action or as an inclination
to move, possessed by the particles of the subtle matter. From
this explanation, comparable with the idea of pressure in a liquid
or of impact amongst panicles in motion, Descartes attempted
to derive the laws of reflection and refraction. The same spirit
was shown in the mechanical explanations offered in the
Meteors : atmospheric phenomena and the rainbow were given
explanations based on known or partly known scientific prin-
ciples. In the Principia Philosophise Descartes dealt, among other
things, with the nature of matter and the general laws of motion.
The whole of this "system" rested on insecure foundations and
there was a temptation to ignore small but " stubborn and irre-
ducible facts " which did not fit in. Since a perfect vacuum was

THE LIFE OF CHRISTIAN HUYGENS 25

something to which Descartes denied existence the Torricellian
space in a barometer tube was supposed to be filled with the
Cartesian subtle matter which, like the ether of the nineteenth
century scientists, penetrated almost everywhere. A crucial ex-
periment was performed in which a sealed and empty bladder was
placed in an evacuated tube. The fact that the bladder expanded
was explained by Huygens as due to a small amount of residual
air: Roberval, an original and controversial writer, also con-
sidered the experiment disproved rather than supported the
Cartesian theory.

Mersenne died in September 1648, but his influence on Huy-
gens had been important. Although he was no great physicist or
mathematician he stimulated the criticism of ideas; he was, for
example, strongly opposed to Descartes's well-known treatment
of animals as automata. In the years between 1648 and 1657 Huy-
gens, from being a youthful admirer of Descartes's philosophy,
became more and more critical. He wrote frequently to his old
teacher Schooten and to the mathematician Slusius about Des-
cartes 's demonstrably false laws of impact between elastic bodies.
The laws, he wrote, did not agree with any experiments and the
fifth law conflicted with the second. Before 1656 he had com-
pleted his own important work on the subject (see p. 109) but
some twelve years elapsed before he communicated his conclu-
sions to contemporary men of science. The complete treatise,
De Motu Corporum ex Percussione, was not published during his
lifetime.

The first published work of Huygens came out, however, as
early as 1651, when he was only twenty-two. This was his Cyclo-
metrise, a treatise written to show up the fallacies of the mathe*
matician Gregory de St. Vincent committed in a book of 1647
where Gregory had claimed to have developed no less than four
different ways of " squaring the circle ". The task of replying to
Huygens's serious objections was left to certain pupils, and
notably to Ainscom. The result was considerable prestige for
Huygens, for he was seen to have proved his case. The larger
work, De Circuit Magnitudine Inventa. which appeared in 1654,
it is safe to say, assured him of a place amongst the leading
mathematicians of the day. He was hailed as the reborn Vieta
and compared with Pappus and Apollonius, two giants of
classical Greek geometry. The comparison was, in fact, not inept.
In the years following 1652 Huygens spent a lot of time on re-

26 THE LIFE OF CHRISTIAN HUYGENS

ducing to algebraic analysis problems which Archimedes, Nicho-
medes, and other Greek mathematicians had been able to solve
only through geometry. Without these early studies it may be
doubted if Huygens could have succeeded in the great problems
he was later to tackle.

Before the death of Mersenne Christian had hopes of going
to Paris in the company of his father, but the idea was post-
poned. In 1649 came the first of two revolts by the nobility
against the rule of Anne of Austria and Mazarin during the
minority of Louis XIV. Until 1653 t " ie situation continued to be
uncertain; twice Mazarin was a fugitive, Anne was hunted from
Paris and the monarchy was in jeopardy. The rebellious nobility
were in league with Spain and the times were not propitious for
the Huygens' visit. Not until 1655 was the long-projected visit
made. The intervening five years were spent chiefly on Huygens's
early researches, interrupted only by another journey to Den-
mark and some time spent in the Low Countries. Huygens's
important work on telescope construction dates from these days.
The first telescopes were made in Holland early in the century,
but they were very imperfect and it is remarkable that observa-
tions such as those of Galileo were ever made with such instru-
ments. The task of improving the telescope occupied Huygens
throughout his life and in this he was encouraged by his father
and had, from time to time, the skilful collaboration of his
elder brother. By means of his own telescopes Huygens
made his important observations on Saturn and a copy of a letter
written at this time bears two rough sketches, one of Jupiter and
the other of Saturn showing appendages. The contents of the
letter do not relate to these matters and the date of the draw-
ings is uncertain. The discovery of Saturn's ring cannot be put
earlier than February or March 1656. Before he went to Paris in
July 1655, however, Huygens had made the interesting discovery
of a satellite of Saturn. The study of this anomalous planet
whose irregular contour was such a mystery was continued by
his brother Constantin in his absence.

The fertility of Huygens's mind at this period was truly
astonishing and it can be matched only by comparison with
Newton. Fundamental research in pure and applied mathema-
tics, optical studies including important work on the theory of
lens systems, the invention of an improved eye-piece for the
astronomical telescope, and to crown the practical side of his

THE LIFE OF CHRISTIAN HUYGENS 27

work, the discovery of Titan, all belong to this period of
his life. Yet there was a curious weakness in this energetic
mind, a flaw implanted perhaps by Descartes's brilliant
philosophizing. For the man of science who was himself
a few months later to discover Saturn's ring seems to have
concluded that, with the discovery of six planets and six satel-
lites, the human mind had reached the limits of the solar system.
This preoccupation with numbers reminds one of Kepler and
shows how persistent were old currents of thought.

In 1655 Louis XIV was only seventeen and France continued
to be governed by Mazarin. The second "Fronde" was at an end
but this disorder and scramble for power left its results, abortive
though it had been. The conditions were precisely those which
could not but impress Louis with the need of becoming master
and his reign from 1661 onwards was characterized by the great-
est absolutism.

For a son of a noted Dutch diplomat and man of letters
young Huygens remained extraordinarily aloof from the turmoil
of events. His habits soon became those of a scholar and at
twenty-six there was a marked vein of seriousness in his pursuits.
During his five months in Paris, however, he revelled in the
opportunities of pursuing the arts as well as the sciences. Music,
the drama, and the society of intellectual and artistic people
made life in the capital extremely interesting. At the country
house of Conrart, the protestant secretary of the French
Academy, he met Jean Chapelain, a mediocre but popular poet
and a man of cultivated tastes, and Marie Perriquet, an attractive
young woman who seems to have shown interest and some ability
in scientific problems. The comic dramatist Scarron, the astron-
omer Boulliau and the philosopher Gassendi were also among his
new acquaintance.

Gassendi was at this time an old man. It is not clear how
much Huygens could have been directly influenced by the philo-
sopher on the occasions when they met, but his indirect influence
on Huygens and several other men of science was considerable.
Gassendi was at that time an important, perhaps the most im-
portant, opponent to Cartesian teachings. His objections, more-
over, reflected the influence of Galileo: logical deductions for

28 THE LIFE OF CHRISTIAN HUYGENS

him were useful only so long as they did not conflict with the
physical facts. For Descartes mathematical and logical deduc-
tions could be valid irrespective of verification from experience.
The fact that this appears to us an impossible and irresponsible
attitude must be attributed to the influence of Gassendi, Huygens
and Leibnitz as well as of Newton in the history of thought.
Even in a more detailed way, however, Gassendi had an im-
portant influence on Huygens. He held an atomic theory which
was later developed by Boyle and with this went a belief in
scientific materialism. To Descartes's " cogito ergo sum " he
objected that existence might be inferred from any other action
besides thinking. Unfortunately Gassendi's name has become
linked, not only with the revival of atomic doctrines, but also
with the doctrine of mechanism. As a matter of fact Gassendi
did not take up this extreme position. The atoms of bodies, he
held, were not eternal or unproduced or moving of their own
accord a problem which he seems to have viewed with the same
perplexity as we feel now for the rotation of nebulae. God, for
Gassendi, was the creator and first cause, He was over and above
the physical world.

From 1653 up to his death in October 1655 Gassendi lived at
the house of Habert de Montmor, a wealthy amateur of the
sciences, who gathered together at his house, 7 rue Vieille du
Temple, many who had formerly met at the cell of Mersenne.
This " Montmorian Academy " was an important forerunner of
the Academic Royale des Sciences. As in London and in Florence,
an informal gathering of men, free from the " systems " of the
universities, committed to no philosophy save that of enquiry,
founded the modern organization of scientific work. Besides en-
quiring into new phenomena something was done to conserve the
past. Gassendi wrote a life of Tycho Brahe and Copernicus. It
is worth noting that he preferred the cosmology of the former to
that of the latter. His pupil, the poet Chapelain, took a keen
interest in the progress of scientific studies and formed a strong
friendship with young Huygens. Chapelain knew little science
or mathematics but his zeal was great and he assisted Sorbi&re,
permanent secretary to the Montmorian Academy, to draw up
its rules and maintain its foreign correspondence. Chapelain's
letters to Huygens, after the latter's return to the Hague, show
that he kept his master's atomic doctrines to the fore and con-
scientiously maintained a critical attitude towards Cartesianism.

THE LIFE OF CHRISTIAN HUYGENS 19

After Gassendi's death in 1655, in fact, his place in Chape-
Iain's life was taken by the newcomer Huygens. Unfortunately,
the death of Gassendi was the beginning of a series of troubles
for Montmor. First he lost his child and then his wife fell ill;
finally, when she was recovering, his sister died. These misfor-
tunes brought to an end, temporarily, the meetings held at his
house. There may have been other reasons, for Gassendi's assist-
ant, de la Poterie, and Huygens disliked each other, and Pierre
Petit and Thevenot were hostile to Sorbiere. Numerous petty
squabbles occurred and marred the work of the " Academy " for
a period. Nor were other societies in Paris more successful.
Thevenot later started discussions at his own house (in 1663) but
these ended in 1664 because of the expense, part of which was
incurred by keeping the mathematician, Bernard Frenicle de
Bessy, and the anatomist, Steno, at his house. Groups supported
for a time by Henri Justel or the Abbe Bourdelot suffered no
better. It was through experience of this sort, as will be seen,
that leading amateurs of science were brought to the conclusion
that the Government should be responsible for the maintenance
of a permanent academy.

At the time when Huygens first visited the Montmor group
there was, as usual, little contact with the Sorbonne. Some of the
university teachers attended meetings, but perhaps as much out
of suspicion as out of sympathy. The ecclesiastical authority of
the university colleges felt itself challenged by Cartesianism, and
Gassendi's views were no more popular. Nevertheless, a rational-
ist sect existed within the Church and this was not wholly
opposed to new ideas in natural philosophy. Among its
supporters the writings of Du Vair and Descartes were in-
creasingly popular. But the support of neither the Jesuits nor
their opponents the Jansenists was of any value for the cause
of science.

Huygens enjoyed the meetings of the Montmor group and
was anxious to prolong his stay in Paris. From the mathemati-
cians he learned of problems on probability which occupied Fer-
mat and Pascal about this time. It is of interest that his own
little treatise on the subject, written after returning to Holland,
became a classic. Before he left France he told Boulliau and
Chapelain of his discovery of Saturn's satellite and the latter
urged on him the importance of publishing the details. Huygens
obviously wished to settle the problem of the ring first; he was

30 THE LIFE OF CHRISTIAN HUYGENS

pleased at the discovery that already his telescopes were as good
as any that were to be had in Paris.

IV

In 1610 Galileo made a number of important telescopic observ-
ations. In January he found that Jupiter had four satellites; in
July he made out the appearance of Saturn as consisting of
"three spheres which almost touch each other, which never
change their relative positions, and are arranged in a row along
the zodiac so that the middle sphere is three times as large as
the others ". In the same year he distinguished separate stars
in the Milky Way and saw the phases of Venus. His work left
the Copernican theory in a much stronger position but certain
unresolved doubts still remained. This " tri-spherical " form of
Saturn for example was something completely anomalous. Were
these outer spheres a peculiar type of moon? On the Copernican
theory it appeared probable that other planets besides Jupiter
would be found to possess satellites.

This is where Huygens's work commenced. At the age of
twenty-six he made a search for satellites of Venus and Mars, but
in vain. Turning his own twelve-foot telescope on Saturn, it
appeared to him much as it did to Galileo. The nature of the
lateral bodies or appendages could not be distinguished. Leaving
this problem on one side, however, Huygens at eight o'clock in
the evening 'of March -25th, 1655, noticed a small star very near
the line passing through the planet and its appendages or
" anses ". His suspicions that this would prove to be a satellite
were strengthened during the following days, for the position of
the star altered. After a few weeks Huygens decided that the
period of the satellite (Titan) was sixteen days and four hours.

This discovery, as has been mentioned, was made before Huy-
gens went to Paris in July. No doubt several of Montmor's group
discussed with him the puzzling problem left by Galileo. Heve-
lius, the noted astromer of Danzig, confirmed the planet's pecu-
liarities, his telescope being not much better than that of Galileo.
Huygens recognized that everything depended on improving the
instrument. The diverging eyepiece of the Galilean telescope
restricts the field of view and Scheiner, about 1630, successfully
made the first instrument having two or three convex lenses.
Telescopes of this type have a better field of view but are more

THE LIFE OF CHRISTIAN HUYGENS 31

subject to the defect of chromatic aberration, a matter not then
understood. Huygens went to the lens maker Mocchi while he
was in Paris and learnt all he could from him. When he returned
to the Hague he worked continuously on lenses, trying amongst
other things to produce a hyperbolic or elliptical surface, but both
proved too difficult. He succeeded, however, in building a larger
telescope having twice the magnifying power of his twelve-foot
instrument and this enabled him to study Saturn more closely.

In the winter months of 165556 great progress was made to-
wards solving the problem. Instead of the " tri-spherical " form
he was able to distinguish a sort of band passing across the
middle of the planet and drew it in the form :

A slightly later drawing of Saturn showed it in the form :

It is difficult to imagine what his conjectures were at this point.
A new twenty-three foot telescope with the best lenses he could
make was assembled with all speed and this was in use after Feb-
ruary ipth, 1656. With this instrument the planet appeared much
more distinctly and he at last made a drawing of it showing it
surrounded by a ring :

The drawings he made show that he was struggling to gain a
clearer image and only by degrees became certain about what he
could see (cf. p. 194).

During an interval between June and October the planet was

32 THE LIFE OF CHRISTIAN HUYGENS

not clearly visible but Huygens already felt confident that
his observations had only one interpretation: Saturn is sur-
rounded by a thin ring of matter slightly inclined to the ecliptic.
This idea was concealed in an anagram published in his De
Satitrni luna abservatio nova, which came out in the spring of
1656. When disentangled the anagram reads "Annulo cingitur,
tenui, piano, nusquam cohaerente, ad eclipticam inclinato," viz. :
" It is encircled by a ring, thin, plane, nowhere attached, inclined
to the ecliptic." From his correspondence it is clear that he was
confident of his conclusions as early as February of that year.

The use of anagrams was common in those days. Huygens
adopted the device so as to give an opportunity for other astron-
omers to bring their own discoveries to the light of day " so that
it may not be said that another has borrowed from us, or we
from him ". The method was superseded with the growth of
scientific periodicals. In this instance, Roberval, Hevelius and
one Hodierna all came forward with their own announcements.
Hevelius alone produced anything of importance. His Disser-
tatio de Natura Saturna Facie (1656) contained in fact a complete
theory for the observed periodicity in the phases of Saturn. He
cannot have seen the planet clearly for he supposed it to be ellip-
soidal and not spherical in form; also there were two appendages
physically attached to its surface.

Roberval put forward the theory that Saturn is surrounded by
a " torrid " zone. From this equatorial zone " exhalations " were
ejected and these were supposed to be transparent except when
present in great quantities. The periodicity in the phases was
ignored. Even more remarkable was Hodierna's account of
Saturn. His theory that the planet had the form of an egg or
plum having two dark patches deserved careful verification, Huy-
gens caustically remarked. Certainly such an appearance called
for study by a better telescope than one of a magnification of
five!

Bouliiau was unable to see the satellite Titan and this made
Huygens suspect the quality of his telescope and for this reason
to trouble little about Boulliau's criticism of the ring theory. It
mattered rather more when Wallis, the English mathematician,
wrote to say that the English had forestalled him. This, however,
proved to be a practical joke by Wallis, who otherwise is known
only for the seriousness of his pursuits. 1 However, the details

1 See notes on Wallis and others (p. 212).

PLATE I

Saturn Reproduced from Huygens's MS.

THE LIFE OF CHRISTIAN HUYGENS 33

of the ring continued to give Huygens, as he said, " no little
trouble ". The difficulty was to fix the interval between the phases
and calculate the future appearance of the planet. At the end of
1657 Huygens was able to inform Boulliau of the confirmation
of his theory. " On the ifth of December I saw Saturn with my
big telescope for the first time after it had passed the sun and was
delighted to find it exactly in the form I had predicted according
to my hypothesis." He went on to say that the ring appeared a
good deal larger since its last occultation " so that now the sky
can be seen through it ".

At the crowded assembly of Mommor's circle Chapelain pre-
sented a detailed account of Huygens's studies of Saturn. The
planet was in all other respects normal: it traversed an orbit
around the sun and its axis of rotation was almost parallel with
that of the earth. The axis was always perpendicular to the
plane of the equatorial ring. The solid and permanent nature
of the ring could be clearly perceived. Twice in thirty years
the sidereal period of the orbit the ring appeared to vanish
since it was viewed edge on. The company was a distinguished
one and general praise was forthcoming for the young astrono-
mer's discovery. Even Roberval paid him a generous tribute
and retracted an earlier suggestion that Huygens was indebted
to him for his ideas. He still maintained, however, that his
own theory was to be preferred. Huygens wrote that the ring
was without doubt a great novelty and one to which " in the rest
of the universe there appears to be no parallel ". In June 1659 his
book Systema Saturnium appeared.

Copies of the little treatise were sent to Paris and to Prince
Leopold de Medici, to whom it was dedicated. The prince was
a great supporter of science and founder of the Accademia del
Cimento. Influenced by Boulliau, who sent him criticisms of
the hypothesis, however, Leopold hesitated to express his opinion
of Huygens's work and it was only after a considerable delay
that he acknowledged its importance. Of the noted astronomers
of the day, Hevelius, Boulliau and Riccioli did not accept Huy-
gens's account of Saturn's ring. To this day no-one seems to have
recognized the importance of Huygens's theory that the ring
would be stable under uniform gravitational attraction assum-
ing mechanical resistance to fracture. He did not state that the
gravitational force kept the ring in rotational stability but he
did suggest that Saturn's gravity extended to the ring.

G

34 THE LIFE OF CHRISTIAN HUYGENS

It was as a comment on Copernicanism that Huygens intended
his book to be read. The nature of gravity was, he insisted, the
same for all the planets. A stationary ring of uniform thickness
would, then, be in equilibrium. No further evidence could well
be expected. There are other interesting matters in the work,
but these will be discussed later. What concerns us here is that
the severest attacks which were made on the Systema Saturnium
were made for religious and not scientific reasons. It is curious
that after a period of tolerance the Catholic Church became
bitterly opposed to Copernicanism in the seventeenth century. In
1615 the Holy Congregation had declared all books of Coper-
nican doctrine to be condemned and prohibited. Chapelain cer-
tainly expected trouble and wrote that it was surprising that the
hypothesis of the movement of the earth was allowed to pass in
Holland. Pre Honori Fabri, a Jesuit, and an astronomer,
Eustachio Divinis, were foremost in their antagonism to Huy-
gens. These critics found it necessary to impugn not the argu-
ments advanced by Huygens but his very observations. This
drew a sharp reply from Huygens in his tract Antidivmis, but
the controversy dragged on until Huygens was well established
in Paris in 1666 and will need a further account later. The name
of Fabri is obscure enough now. Nevertheless, under the name
of his friend and pupil, Mousnier, appeared one or two inter-
esting attempts to develop mechanics. The trouble was that
Fabri possessed the outlook of an Aristotelian, for he wished to
deduce the mathematical laws of dynamics from principles of
natural philosophy. So deeply rooted was this habit of mind
at this time that considerable feeling was frequently roused by
ideas which rested on an entirely different attitude. In France
the authorities of the theological college of Paris University tried
to get decrees issued in defence of Aristotle's philosophy and
against the new heresies as late as 167 1. This ridiculous situation
was treated to a sarcastic burlesque by the playwright Boileau,
who thereby did much to wreck the scheme. Later, a more eclectic
outlook existed; Cartesianism entered the Sorbonne itself.

Between 1655 and 1660 Huygens spent much time on the
invention of an accurate pendulum clock. The significance of

THE LIFE OF CHRISTIAN HUYGEN3 35

this invention in the history of science is that it marked the new
interest taken in time as a dimension. We shall see that Huygens
effectually began the study of dynamics. One reason why the
history of mechanics up to his time was really the study of
statics was undoubtedly the tendency resulting from the neo-
Platonic revival of the sixteenth century which coincided with
the decline in Aristotelianism. This tendency was to reduce most
physical problems to geometry. But the absence of accurate time
measurement was undoubtedly another reason. Galileo, it will
be remembered, used a water clock in his experiments on acceler-
ation over the inclined plane.

Very probably it was in the first place Huygens's early enthu-
siasm for astronomy which led him to tackle the problem of the
pendulum clock. Balance clocks existed, of course, from much
earlier times, probably from the thirteenth century, but they
were crude and unreliable machines. Tycho Brahe used one in
conjunction with his mural quadrant and corrected for its errors
by comparison with the sun. The measurement of the time of
passage of a star across the meridian could be used to replace the
measurement of its meridian altitude, this being a more difficult
measurement and rendered uncertain through the absence of
reliable corrections for the atmospheric refraction. Also, as a
member of a seafaring nation, Huygens could not have failed
to know that an accurate clock would afford the simplest method
of determining longitudes at sea. This question seems to have
interested him more after he had made his first clock.

His first publication, Horologium, a short treatise describing
the application of the pendulum to the escapement, appeared
in 1658 but the invention was known to his friends some two
years earlier. Unfortunately a controversy arose through the
claim made by Leopold de Medici that the priority for the in-
vention belonged to Galileo. Roberval and a Paris clockmaker,
Thuret, also claimed that they had anticipated Huygens. The
whole history of the pendulum clock has in fact been obscured
by various energetic contestants.

In 1598 the King of Spain offered a prize of one thousand
crowns for a means of finding longitudes at sea and this was
followed by an offer of ten thousand florins by the States General
of the Netherlands. Now Galileo is said to have discovered the
approximate isochronism of a simple pendulum in 158 1 . He him-
self, in 1636, offered to the States General a method of determin-

36 THE LIFE OF CHRISTIAN HUYGENS

ing longitudes based on the telescopic observation of the
occultations of the moons of Jupiter. It was proposed to publish
an almanack of the eclipses of these moons and to use a
" numeratore del tempo " to measure the time intervals. This
instrument, from all accounts, was merely a simple pendulum
maintained swinging by hand and fitted with a completely im-
practicable mechanism for counting the swings. Admiral Read's
committee did well to reject the " invention ". It is quite possible
that in 1637 Galileo came across Leonardo da Vinci's drawing
for a clock regulated by a pendulum. It was in this year that
da Vinci's manuscripts were given to the Biblioteca Ambrosiana
at Milan by Galeas Arconati and the donor is known to have
been at pains to bring his treasure to the notice of contemporary
men of science. Galileo in this year became blind after a long
period in which his eyes were diseased but he had around him
Viviani, Torricelli and his son Vincenzio. Viviani, writing to Leo-
pold in 1659, described from memory how Galileo discussed with
his son the construction of a pendulum clock. The date given
was 1641. Whether Vincenzio ever completed its construction
is not known. It is certain that Huygens knew nothing about
the design until after the publication of his Horologium in
1658. A copy of this was sent to Leopold de Mediei, who replied
guardedly, pointing out that Galileo had had the same idea.

As against the theory that Galileo was indebted in any way
to da Vinci it needs to be mentioned that the design commonly
attributed to Galileo differs somewhat from that shown (though
rather imperfectly) in da Vinci's note-books. Huygens's design
differs from that of Galileo and was, in fact, closer in prin-
ciple to that of da Vinci.

It was in the records of the Accademia del Cimento of 1662
(published in 1667) that the implication of Huygens's plagiarism
was really blazoned abroad. Here it was stated that Vincenzio
had put his father's design into practice in 1649. No details were
given and the illustration simply showed a drum-shaped clock
mounted horizontally on a vertical pedestal. From the under-
neath side of the drum hung something resembling a simple
pendulum, the mode of attachment of which had to be
imagined. It is obvious that a simple pendulum would be useless,
since it is impossible to give an impulse to the thread and pre-
sumably a thin iron rod was intended. Nevertheless, Matteo
Campani stated that he saw the clock constructed by Vincenzio

THE LIFE OF CHRISTIAN HUYGENS 37

" an antique and rusty machine not at all complete " and
Leopold's letters certainly suggest that such a clock was in
existence, though whether it conformed to the diagram is not
known. The clock was never forthcoming and it has generally
been supposed that Viviani pressed the whole case for Galileo out
of a desire to honour his master. Nevertheless, the evidence does
seem to indicate that Galileo did precede Huygens in achieving
the successful application of the pendulum to the escapement,
but that the complete clock was constructed is exceedingly doubt-
ful. Probably da Vinci was the first to have the idea and still
more probably Huygens was the first to carry it through to
fruition. In a later chapter it will be shown how astonishingly
thorough in every particular Huygens's work was; so completely
did he clear up the theoretical and practical problems that he is
in a real sense the father of modern time-measurement. Samuel
Coster, his clock-maker at the Hague, made a large number of
clocks to his design and these were the first to be commercially
available.

As has been mentioned, Huygens probably interested him-
self in the longitude problem after he had made his first clock.
He may have read the work Nieuwe Geographische Onder-
wijsinge in Dutch by Metius (1614) which pointed out that it was
only the irregularity of balance clocks which prevented them
from supplying a means of finding longitudes at sea, but in any
case the relation between local time, standard time and longitude
was well known. The pendulum clock was a far better instrument
but it was very easily disturbed. Huygens consistently under-
estimated this problem of the movement of the ship. He
thought that a pendulum whose period was independent of the
amplitude of swing would enable this difficulty to be overcome.
The idea was ingenious and led to the discovery that the period
of oscillation of a cycloidal pendulum is independent of the
amplitude, but the practical value of the discovery was strictly
limited. A great deal of time was occupied by the investigation
of this problem, conducted as the research was by elementary
and tedious mathematical methods. In practice a cycloidal
pendulum may be constructed by allowing a simple pendulum to
swing between two curved metal plates along which the thread
curves itself on each half of its swing. In applying the idea to
the clock pendulum Huygens used a short ribbon attached to a
rigid pendulum. These plates or " cheeks " were first tried in

38 THE LIFE OF CHRISTIAN HUYGENS

1657 or during the last days of 1656. Towards the end of 1659
Huygens showed that theoretically they should themselves
possess the form of cycloid arcs. He was immensely pleased
with this discovery and ranked the geometrical part of
the work above all the rest.

About this time (1659) more detailed accounts of
Galileo's escapement became available. Models of this
escapement have since been made and it cannot be
said to work satisfactorily. As Huygens pointed out
at the time this escapement imparts a very uneven
movement to the pendulum. His own clock remained
therefore the only one in this field.

Continued efforts by other inventors, including the
clock-maker Thuret, to profit from the invention
drove him to the unwelcome decision that he should
take out a patent or " privilege " to protect his rights.
Much delay occurred before the French " privilege "
was issued, but thereafter Huygens's priority was
recognized and he made some profit from the construction of
clocks to his design.

The story of the clock needs to be told with reference to
certain of Huygens's mathematical researches. His first essay in
this field had dealt (1651) with some fallacious work by Gregory
de Saint Vincent on the rectification (or measurement) of certain
curved lines. Huygens became interested in the rectification of
curved lines known as conies and in the age-long problem of the
rectification of the circle. When Boulliau sent him some problems
by Pascal on the curve known as the cycloid in 1658, therefore,
the subject was by no means a new one. These problems, to which
Pascal had already obtained solutions, and which he set for the
interest or exasperation of other mathematicians, were known
as the " Dettonville " problems, this being the pseudonym under
which they were issued. Huygens succeeded in solving some of
the necessary preliminary problems but found the main ones so
difficult that he declared himself unconvinced that they had ever
been solved. When he later came across a rectification of the
cycloid by Christopher Wren he expressed his admiration. It
was, he commented, the first curved line known to be rectified,
and he wondered if it were the only one which could be rectified.
There was some correspondence between Huygens and Pascal
on the " Dettonville " problems. Pascal praised Huygens's

R|,4

m

One

THE LIFE OF CHRISTIAN HUYGENS 39

penduhim clock very highly, but such was Huygens's esteem for
Pascal as a mathematician that he deprecated such mechanical
inventions. There was, he remarked, little science or subtlety in
such things. About this time Pascal's adherence to the Jansenist
sect was somewhat weakened but his periods of religious pre-
occupation invariably interrupted his most interesting work
and his most promising friendships. So it was in his relations with
Huygens. The latter was eager to collaborate but closer relations
were frustrated.

It is clear that at some time between Septetnber 1659 anc *
January 1660 Huygens discovered the theoretical form of the
small plates or " cheeks " for his corrected pendulum. These
dates may be fixed by an examination of his correspondence.
From this it seems that he did not at first use the metal plates
except for clocks in which a large swing was employed. Later,
taking the view that a marine clock would benefit from having a
pendulum swinging through a large arc, he felt that it was im-
perative to discover the theoretical form of the restraining plates.
His success in this problem gave him the pleasure of a mathe-
matician with a pretty solution. He announced that the second
edition of his Horologium would contain "a fine invention
which I added to the clock a little while ago ". It appears prob-
able that, although Pascal's problems were of a very different
character, the interest of the cycloid led Huygens to make his
investigations.

The improved clock was in use towards the end of 1669 and
was adopted after that date as being the best time measurer then
made. Earlier astronomers, notably the Landgrave of Hesse
(who used balance clocks made by Byrgius in the sixteenth
century), Tycho Brahe, and later, Hevelius and Mouton, recog-
nized the importance of time measurements, but Roemer
and Flamsteed, late in the seventeenth century, were really the
first to use the clock systematically. Delambre, in his great
Histoire de rAstronomie Moderne, states that Huygens " started
the great revolution " in practical astronomy by the invention of
the pendulum clock.

Curiously enough the cycloidal pendulum, in spite of its
elegance, did not have a very long life. It was recognized by
Huygens himself that small circular arcs were equally accurate
and in his model of 1658 he was able to restrict the size of swing.
After the application of the anchor escapement by a London

40 THE LIFE OF CHRISTIAN HUYGENS

clock-maker, Clement, in 1680, few cycloidal pendulum clocks
appear to have been made. The anchor escapement causes the
pendulum to describe small arcs of constant amplitude and this
made the cycloidal pendulum for all but marine clocks super-
fluous.

As for the marine clock or chronometer, although Huygens
constantly considered himself near to success, it proved eventu-
ally to be a failure. The pendulum seemed for many years to be
the only means of controlling the going of the clock with
sufficient accuracy. Huygens accordingly tried various forms of
suspension and various forms of pendulum, all designed to with-
stand the movement of the ship, but none proved to be a practical
proposition. Such was the commercial rivalry of the various East
India Companies, however, that he was encouraged to persevere,
and persevere he did up to the last year of his life. How near
he came to success will be described later.

Meanwhile the astronomers obtained their longitudes by
Galileo's method of observing the recurrent eclipses of the
satellites of Jupiter. Cassini, working at Paris, drew up the first
tables for the observation of these satellites and, with Richer, in
consequence of this work was able to make the first modern
estimate of the distance of Mars. But astronomical methods
were clearly unsuited to the determination of longitudes at
sea.

Huygens's great essay, Horologium Oscillatorium, on the
construction of the clock and all the relevant propositions on the
cycloidal pendulum and on the centre of oscillation did not
appear until 1673. Already, however, he had made notes for the
work and even wrote to Chapelain in September 1660: "The
treatise on the clock has been finished a long time but tnere is
no means of having it printed before my journey ..." This
refers to an extended edition of the original Horologium which
included a treatment of the cycloid; much was yet to be added
before the work reached its final form. In October Huygens left
the Hague for Paris.

VI

In considering the encouragement given to literature and the
arts in France during the seventeenth century, the credit belongs
almost as much to Mazarin as it does to his successor Colbert.

THE LIFE OF CHRISTIAN HUYGENS 41

Mazarin it was who gave pensions to many of the great writers
who made this the golden age of French literature. Molifcre,
Balzac, Descartes, Pascal, Racine, Corneille, Boileau and others
benefited from Mazarin's patronage and his example was
followed by Colbert after 1661.

Nevertheless, the establishment of the Academic Royale des
Sciences would never have been achieved if the men of science
had waited for Colbert. Up to 1663 what progress there was in
any regular pursuit of science was made by men who were
associated with one or other of the amateur societies, Montmor's
and Thevenot's being much the most important. But in this
year there was a rather defeatist air about the correspondence on
the subject of a permanent academy and Sorbiere summarized
the difficulties in an account sent to Colbert. There had to be
appeals by Sorbtere, Thevenot and the Abbe d'Aubignac, how-
ever, before any impression was made. Finally Auzout publicly
appealed to the king's pride (and vanity) and after the Peace of
the Pyrenees things began to look more hopeful.

The scientific societies did indeed develop under difficult con-
ditions. In England, for example, where the Royal Society was
taking shape, there was in progress a stern struggle between king
and parliament, a deep religious dissension and in the com-
mercial sphere a rivalry with the Dutch which had become acute.
In Holland, on the other hand, it was understood that the chief
political problem of the time was the neutralizing of the grow-
ing power of Louis XIV. Civilization was passing through a
critical period and internal dissension as much as external danger
made the times, one would have thought, unpropitious in most
countries of Europe for the growth of societies with the calm in-
terests of natural science as their pursuit. But perhaps these
interests were all the greater attraction; as Sprat wrote after-
wards, the members of the Royal Society wished simply for " the
satisfaction of breathing a freer air, and of conversing in quiet
with one another, without being engaged in the passions and
madness of that dismal age ". Needless to say their work was not
always taken seriously and both in Paris and London there were
scoffers who doubted the worth if they did not mistrust the in-
fluence of " natural philosophy ". Pepys recorded that Charles II
" mightily laughed at Gresham College for spending time only
in weighing of ayre and doing nothing else since they sat ". But
Colbert certainly saw that there was more to those pursuits than

42 THE LIFE OF CHRISTIAN HUYGENS

met the eye. It may be doubted if Louis XIV, unaided, saw any-
thing significant at all in what was going on.

In spite of the difficulties an extensive correspondence was
carried on between the men of science. Paris was at first the
chief centre of experiment, but London later rivalled and then
surpassed it in activity. The meetings of the scientists seem to
have been devoid of political motive; religious difference only
rarely caused antipathy, national differences scarcely ever. As
will be seen, Huygens spent many years in Paris under the
patronage of Louis XIV, even though his family had a long
association with the house of Orange and, after 1672, the young
Prince of Orange headed the resistance to French invasion. It
might have been thought that Huygens, a protestant Dutchman,
would have been regarded as a spy but this was not the case. Not
until 1683 did it become clear that, with Colbert's death, support
for his continuance in Paris was gone. Colbert was succeeded by
Louvois, and in 1685 the Revocation of the Edict of Nantes
caused many protestants to leave the country. All this, however,
lay in the future.

In 1658 Montmor charged Sorbiere with the task of drawing
up rules for the meetings of the assembly which were held
regularly at his house. The keynote of the rules, in the form
finally adopted, was the need of restricting " the vain exercise of
the mind in useless subtleties ". Mere philosophizing, it was
agreed, was profitless. Unfortunately the assembly did not
appreciate how their aim should be achieved : without a pro-
gramme of experimental work directed to the solving of selected
problems too many of their meetings continued to degenerate
into philosophical combats.

Huygens returned to this gathering of scientific amateurs in
1660 and was introduced by Chapelain on November and. At
Montmor's, he wrote to his brother, " there is a meeting every
Tuesday where twenty or thirty illustrious men are found
together. I never fail to go ... I have also been occasionally to
the house of M. Rohault, who expounds the philosophy of M.
Descartes and does very fine experiments with good reasoning
on them ..." Rohault's meetings were held on Wednesdays
and began about 1658. There is no doubt that he did a great deal
to make science popular in Paris. Unauthorized editions of his
lectures were published, so great was the popular interest. It is
not surprising that Huygens approved of him, although no

THE LIFE OF CHRISTIAN HUYGENS 43

association between them seems to have occurred. Rohault was
enlightened and modern in his attitude but he was an expositor
and lecturer rather than an original thinker. Where he abandoned
Aristotle he followed Descartes.

At Montmor's house Huygens noted a room "full of
beautiful paintings ", a cabinet of curious inventions and mathe-
matical instruments, and drawings by Albert Durer. With the
astronomers he discussed his work on Saturn and the problems
of lens grinding; with the mathematicians, as he noted, " my
theories of the superficies of conoids and spheroids and the new
properties of the cycloid for pendulums "; and with the clock-
makers and telescope-makers Huygens also passed interesting
hours. He met Conrart, Roberval, de Carcavy, Pascal, Pierre Petit,
Sorbiere, Desargues and others. Some of these names will recur
later. 1 He corresponded with Fermat and with his friend Boulliau,
then staying with Hevelius at Dantzig. With Robert Moray,
a prominent member of the London group of " scientists ",
he also began a correspondence. These men were all really
amateurs and the title astronomer, in most cases, for example,
simply indicates the kind of work for which a particular man
showed especial interest and in which he spent his leisure time.
Nevertheless, a man like Cassini, later invited to work at Paris,
represents the new type of professional worker in that most of his
time was in fact spent on genuine systematic work. Huygens
also belongs to this class. Real specialization in the modern sense
was quite unknown, of course. Like all the early scientific assem-
blies of the mid-seventeenth century the Montmorian society
cast its net almost too wide. Huygens's diary records dissections
of human bodies, the examination of machines for which per-
petual motion was claimed, the making of lenses and telescopes
and many other matters. Too much time, Huygens considered,
was spent in arguments of a purely philosophical nature. He
felt that a sterner discipline, a greater application, was needed
than could come out of the performance of merely curious ex-
periments and the holding of discussions.

Nevertheless it must have been an interesting and stimulat-
ing experience to have met so many natural philosophers, all of
whom felt the common interest in the new study of nature. At
the house of the Due de Roannes, Huygens, in December 1660,
met Pascal. Eight days later the Duke, with Pascal, visited

1 Sec notes on Persons Mentioned, pp. 212-6.

44 THE LIFE OF CHRISTIAN HUYGENS

Huygens at his lodgings in rue Sainte Marguerite and, wrote
Huygens, "... we talked of the force of water rarefied in
cannons and of flying; I showed them my telescopes." Pascal was
at this time a sick man. The writer of Provincial Letters had, in
fact, by this date retired more or less completely from the world.
Less than a year later he was dead.

The publication at this time of tracts against Huygens's
account of Saturn shows that the orthodox Jesuits were not pre-
pared to ignore the author's Copernican doctrines. Pere Fabri
especially opposed what he called Huygens's "furtive insinua-
tion" of the Copernican "error". In his Brevis Annotatio in
Sy sterna Saturnium Christiani Eugenii (1660) he presented his
own fantastic theory, although this work was published over the
name of the astronomer Divinis. According to this theory the
planet had two luminous bodies (lucidi) and three dark ones
(obscuri) placed around it and the different relative positions of
these bodies were the cause of its observed phases. This criticism
drove Huygens to compose his Brevis Assertio Systematis
Saturnii within the year. Hevelius is said to have been so favour-
able to this reply that he abandoned his own theory in favour of
the theory of the ring. Leopold, to whom both Divinis and
Huygens dedicated their publications, remained reserved. In
1 66 1 he sent Huygens a further pamphlet by Divinis and Fabri
but it did not seem to Huygens to deserve a reply. It is note-
worthy that by January 1665 even Fabri recognized the truth of
the ring theory, convinced at last by the excellent telescopes of
Guiseppe Campani. Huygens was extremely pleased by this
conversion of his critic. "No-one, 5n my opinion, could reason-
ably reproach me " he wrote, " for having adapted my account
of Saturn to the system of Copernicus . . . the truth of the
matter can only be explained by following Copernicus, and
indeed our system of Saturn corroborates his own strongly."

VII

One of the reasons for Huygens's visit to London in 1661 was
undoubtedly his desire to obtain information of the society of
men of science then meeting at Gresham College, the society
which by charter became in 1662 the Royal Society. He arrived
in London in March just before the coronation of Charles II
and left for the Hague at the end of May.

THE LIFE OF CHRISTIAN HUYGENS 45

The London which Huygens saw in 1661 was the London
which was largely swept away by the Fire and he was not at all
favourably impressed with the condition of the town. All, he
found, was in marked contrast with Paris : the smoke from the
furnaces of the brewers, soap boilers and dyers; and the stench
of narrow alleys innocent of drainage and sanitation. Even
Gresham College had been rendered malodorous. Monk's sol-
diers had for a year used it as a barracks and it was, Bishop
Wren wrote to a member of the society, " in such a nasty con-
dition, so defiled, and smells so infernal, that if you should now
come to make use of your tube [telescope] , it would be like Dives
looking out of hell into heaven." This was in 1658 or the " fatal
year 1659 ". With the " wonderful pacific year 1660 " meetings
of the " invisible college " as Boyle called it, recommenced. By
1 66 1 the college was presumably cleaned up. Huygens, at any
rate, had only admiration for the proceedings there. Brouncker,
Moray, Oldenburg, Godard, Boyle, Wallis and many others were
familiar figures at the meetings of the society and their activities
seemed to him to surpass anything done in Paris. The observa-
tion of stars was done in the garden of Whitehall Palace and
there Huygens tried his own telescope lenses, sent over by his
brother Constantin. These proved to be better than the English.
The Duke and Duchess of York came out to observe the Moon
and Saturn.

Huygens's meeting with Wallis is of especial interest. This
great mathematician showed in his Mechanica sive de Motu
(1669-71) that he had much to contribute to mechanics. The his-
torian Duhem has given the opinion that this work was " the
most complete and the most systematic which had been written
since the time of Stevin ". In his work Wallis generalized the
idea of force which up to his day was used only in connection
with gravity. Huygens's English was at this time not at all good
but he saw that it would be most valuable to keep in communi-
cation with Wallis, as indeed with others of the Gresham College
group. This was the begining of a life-long association with the
English men of science. It is striking that, in spite of the official
position Huygens came to have in the Academic Royale des
Sciences, in 1670, when he feared that he had only a short time
to live, he made arrangements to entrust his papers not to the
Paris society but to members of the Royal Society. Through
Oldenburg, the indefatigable secretary of the Royal Society, Huy-

46 THE LIFE OF CHRISTIAN HtJYGENS

gens was fortunately able to remain in close contact with the pro-
gress of science in England.

There is good evidence that the outlook characteristic of the
English men of science was less complicated by the consider-
ations of a priori philosophies than that of the Paris group.
Francis Bacon has, probably, always influenced literary men more
than he has the men of science, but there is no denying his
importance. Bacon was no scientist and his scientific "method "
was the literary man's conception of science. He never advanced
as far as Descartes into scientific studies. He foresaw, " he cast
forth brilliant intuitions "; ridiculing Aristotle's natural philo-
sophy he pointed to experiment and observations as the only
means of discovering truth : " Nature to be commanded must be
obeyed." With this spirit the English men of science, neverthe-
less, were thoroughly imbued. Their opposition to Hobbes
illustrated thir belief in empiricism. Hobbes's dictum " Experi-
ence concludeth nothing universally " appeared to them mere
philosophic wind; his excursions into physical science the sort of
thing against which their motto Nullius in Verba was later
aimed. It has to be admitted, of course, that most of the Eng-
lish men of science were shocked by Hobbes's acceptance of the
Epicurean philosophy. This appeared to them to be an approach
to Nature which was neither scientific nor pious. Although he
was not explicit about it, it was the more empirical attitude of
the English men of science which impressed Huygens so favour-
ably.

Before leaving London Huygens took part in a determination
of the comparative sizes of the ring and globe of Saturn. He was
pleased to find that his account of the planet was accepted with
admiration. Huygens also saw a transit of Mercury from Long
Acre, using one of Reeve's excellent telescopes.

After Huygens's return to the Hague his father and younger
brother went on a diplomatic mission to Paris. The elder Con-
stantin Huygens, a man of European repute, then made the
acquaintance of some of his son's associates. Through him the
Montmor group heard of Christian's latest experiments. The
Dutch diplomat took the opportunity to present Louis XIV with
one of his son's pendulum clocks. This gift was opportune, for it
was at this time that Colbert was drawing up his schemes to
excel all past achievements in making Paris the cultural capital
of the world and Louis the pre-eminent monarch of the age.

THE LIFE OF CHRISTIAN HUYGENS 47

Louis, of course, became surrounded by an almost ridiculous cult
which sought to elevate him above everyday existence, but even
this had its merits ! The singling out of writers, poets and men of
science was at any rate one of the better consequences. These were
given rewards totalling many thousands of pounds and the inven-
tor of the pendulum clock later came in for suitable appreciation.
More important, however, was Huygens's subsequent invitation
to Paris to assist in organizing a scientific society under royal
patronage. There were delays in carrying out this project, which
must be ranked high among Colbert's achievements, but in 1666
the societies which had been associated with the names of
Montmor, Thevenot and others received this formal recognition
of the importance of their work.

VIII

The early scientific societies exhibited an enthusiasm and
universal interest which scarcely characterizes the professional
societies into which they have developed. Specialization was vir-
tually unknown and through Latin the members had a means of
communication with foreign societies and with a learned world
which had existed before the new studies had begun. It is not
surprising that men like Huygens were acquainted with the
works of some of the Greek writers, nor that it was the Greeks of
the Alexandrian period that held the greatest attraction. Huy-
gens was only following in the steps of Galileo when he studied
the works of Archimedes, for they contained some of the funda-
mental ideas used in mathematics and in statics. But it was clear
that new ideas of a fundamental kind were needed in mechanics
and it was equally necessary to clear away many plausible
suppositions which had no basis in fact.

Huygens saw clearly that such simple machines as the lever,
the pulley, and the wheel and axle, although thfcy gave a
mechanical advantage, could in no way increase the energy avail-
able. Machines for flying, for propelling boats by means of
springs connected with trains of gears, strengthened his convic-
tion that their limitations resulted from a simple mathematical
identity of some kind. But it was many years before this idea
could be expressed satisfactorily. Desargues seems to have tried
to evolve a proof that perpetual motion is impossible but, proof
or no proof, the impossibility was accepted as axiomatic within

48 THE LIFE OF CHRISTIAN HUYGENS

the realm of mechanics by Huygens. The search for a form of
perpetual motion did in mechanics the sort of work that in
chemistry was produced by the search for the philosopher's
stone. In 1659 a book entitled Mechanica Hydraulico-Pneumatica^
by the Jesuit Schottus, reached Huygens. It was partly about
perpetual motion but it also described Guericke's invention of
the simple vacuum pump. In 1661, while in England, Huygens
saw experiments performed at Gresham College using Boyle's
pump, which was a great improvement on that of Guericke. After
reading Boyle's book, New Experiments Physico-Mechanicall
touching the Spring of the Air (1660) he had a copy of Boyle's
pump constructed in November 1661. It is clear from his corre-
spondence that he repeated many of Boyle's experiments, observ-
ing for himself the boiling of water under reduced pressure, the
absence of propagation of sound and the expiration of small
birds in a vacuum. An important original discovery made during
this work was that of the tensile strength of liquids, an effect
which at that time baffled explanation and which led Huygens
to make far-reaching conclusions on the existence of a subtle fluid
or ether which later came into his theory of light.

The question constantly in view behind all work with
vacuum pumps was whether a complete vacuum could really
exist. Many scientists felt, with Hobbes, that empty space is " an
imaginary space indeed ". A fundamental experiment was to fill
a tube with water and invert it so that the open end was under
water in an open vessel and then to place the apparatus in the
receiver of the air pump. When the pressure was reduced the
liquid fell inside the tube and with continued pumping was
brought down to the level of the water in the vessel. With rather
more difficulty the same result was obtained, approximately,
using mercury in place of water. These results agreed well with
Pascal's explanation of the barometer. However, the appearance
of small air bubbles in water which was placed in the receiver
of the pump raised serious doubts, for the descent of the water
and mercury might be attributed to the dilatation of these
bubbles and not to the evacuation of the upper space in the tube.
Huygens, much to his astonishment, found that if air-free water
was used no descent occurred. If a very small bubble of air was
introduced the descent took place. Boyle's law, however, showed
that the magnitude of the effect was far too great to be
accounted for on the dilatation theory. Huygens's observation

THE LIFE OF CHRISTIAN HUYGENS 49

was confirmed in England, and, at Boyle's suggestion, the effect
was obtained without the use of a pump at all Long barometer
tubes of mercury were inverted and were found by Brouncker to
give the effect if air bubbles were carefully excluded. A column
of mercury 75 inches long failed to descend unless a minute air
bubble was present and then the level fell to the normal position
of about 30 inches.

For several years no explanation of this effect satisfied Huy-
gens. But in 1668 he concluded that there must be a subtle fluid
capable of penetrating glass where the contact of the liquid is
not complete and that the height of the barometer is due to the
combined pressures of this fluid and air. Wallis pointed out that
if the subtle fluid were capable of penetrating glass it would
penetrate the Torricellian space also. It is surprising if Huy-
gens did not see the force of this criticism. Although he saw
an analogy with the cohesion of two wet glass plates he missed
the true explanation, which is that films of moisture (such as
exist on mercury and glass) have considerable tensile strength.
In his Traite de la Lumiere, written in 1678 and published in
1690, much was made of this ethereal fluid in explaining refrac-
tion. In fact, very great importance must be attached to Huy-
gens's experiments with the vacuum pump, for his conclusions
profoundly affected his whole outlook. He became an admirer
of Boyle, whom he supported against the criticism of Hobbes
and Linus. The former, he saw, contributed nothing to natural
philosophy; of the obscure ideas of Linus (Francis Hall) he
thought just as little. Boyle's Skeptical Chymist was published
in September 1661 and Oldenburg gave Huygens an account of
its contents. Later Huygens received a copy of the book which
he read with " great pleasure ". " It contains an infinity of use-
ful and remarkable things," he commented, " and in my opinion
it is worth twenty of these other books which are continually
printed on the matters of Philosophy and Chemistry. This
Carneades certainly speaks very truly, reasons acutely, and with-
out doubt shows the true way to discover the truth of things. ..."
In 1662 Huygens heard of Boyle's famous experiments on the
alteration of the volume of a gas with the pressure and he
read Boyle's retort to Hobbes and Linus (A Defence of the Doc-
trine touching the Spring and Weight of the Air). The kinetic
theory of the gaseous state which originates with this work was
propounded by Hooke as well as by Boyle. Hooke spoke of his
D

50 THE LIFE OF CHRISTIAN HUYGENS

theory as Epicurean after the Greek philosopher who, with
Democritus, expounded an atomic doctrine. With the resuscita-
tion of this doctrine in Europe Gassendi had a good deal to do.
Hooke went into greater detail. The particles of air, he sug-
gested, have " much the shape of a watch spring, or a coyle of
wire " which, having rotatory motion, sweeps out a " potential
sphere ", the volume of which varies with the closeness of the
adjacent particles. Huygens was somewhat uncertain if this
theory was in accord with the fact that at high pressures air
retains its fluidity.

If Huygens gained an interest in experiments employing the
air pump from his visit to London in 1661, the English philoso-
phers gained for their part just as much although in a different
direction. For Huygens, by 1661, had discovered the use of a
particular axiom in mechanics which enabled him to solve
problems which Wren, Wallis and others found especially
difficult. This axiom is a simple one : the centre of gravity of a
system of bodies cannot rise as a result of any motion of the
bodies under gravity. Experiments on the ballistic pendulum,
carried out in Huygens 's rooms in London, showed that he could
by this means calculate the heights to which elastic pendulum
bobs would ascend after collision. Another discovery which
greatly intrigued English mathematicians was the theorem that
oscillations of a body in cycloidal arc, occurring under gravity,
are truly isochronous. At the time he was in London Huygens
had not worked out a complete proof although his note-books
show that the work was well advanced. Many mathematicians
were consequently attracted to the problem in the hope of being
first to provide a proof. Brouncker and Auzout both failed, the
former ignominiously, in attempting this problem, which is
difficult by the old geometrical methods but simple when treated
by means of the differential calculus.

All of this work, which was of first-rate importance, was held
up because of Huygens's attempts to construct a successful
marine clock a task which was obstructed more by the limita-
tions of the artisan's resources than by theoretical difficulties.
Alexander Bruce, Earl of Kincardine, then living at the Hague,
collaborated with Huygens in this work. In January 1663 Bruce
crossed to England with two pendulum clocks suspended from
ball and socket anchorages in the ceiling of his cabin. The
weather was so rough that one clock fell from its suspension and

THE LIFE OF CHRISTIAN HUYGENS 51

the other stopped also. In April of this year two similar clocks
were taken on a voyage to Lisbon by Captain Holmes. One clock
went fairly regularly, and the report on its behaviour is preserved
in the British Museum. A filibustering expedition in 1664 to
the west coast of Africa and Guinea gave Holmes another
opportunity. On one occasion the clocks proved more accurate
than the method of dead reckoning in use. Huygens was
optimistic as a result of this report and quoted it in his
Horologium Oscillatorium of 1673. In 1665 war broke out
between Holland and Britain and this ended English collabora-
tion.

Huygens rather resembles Hooke in the variety of his
scientific interests, but he was far more thorough than his
English contemporary and was besides a more " mathematical
head ". Besides his work on the marine clock and in mechanics,
his work on telescopes and the theory of optics was kept up. It
is worth noticing, this dual activity experimental and mathe-
matical. Huygens used it to obtain a guiding idea rather than
a quantitative result. Very great obstacles then lay in the way
of exact quantitative work except in Astronomy. Huygens thus
knew experimentally what order of aperture was needed, in a
telescope of given length and magnifying power, to produce a
clear and sufficiently bright image. He saw that the building of
longer and yet longer telescopes required improved methods of
lens grinding so as to secure sufficient aperture. It also raised
problems of a purely structural kind. Wooden tubes, suitably
braced, were used for telescopes of about ao or 30 feet; for great
lengths Huygens proposed using two short tubes, one at the
objective and one at the eyepiece and with rings placed along
the intervening space. This method was tried in Paris, but it was
found to be very difficult to align the two lenses as can be
imagined. These " aerial telescopes " gave high magnification
but poor definition. Many astronomers accordingly experi-
mented on the grinding of lenses to forms suggested by
Descartes. Many abortive attempts were made before the idea
was abandoned. Huygens made some use of a machine for
grinding lenses but it was not possible to make really large lenses
by any method then known. His superior knowledge enabled
him to see that the eyepiece could in certain ways be made
to compensate for the defects of the objective. In 1662 he spoke
of using two oculars instead of one as a " new manner " of en-

5$ THE LIFE OF CHRISTIAN HUYGENS

larging the field of view. The date of his well-known eyepiece
is, however, not quite certain, but it probably was not invented
much before 1662 and it may have been as late at 1666.

Huygens's theory of Saturn's phases was at this time so widely
accepted, and was confirmed by observation as improved
telescopes came more and more into use, that only more detailed
matters remained to be settled. It was questioned, for example, if
the periods of the phases agreed with the theory that the ring
remained at a constant inclination to the ecliptic. Huygens
showed how the phases could be calculated and succeeded in
converting most of his critics. Wren wrote that " when . . . the
Hypothesis of Huygens was sent over in writing, I confesse I was
so fond of the neatness of it and the naturall simplicity of the
contrivance, agreeing so well with the Physicall causes of the
heavenly bodies that I loved the invention beyond my owne ..."
And the accumulation of observations bore gradual witness to
the success of Huygens's work.

Wren's hypothesis is now forgotten. He and Neile, in 1658,
tried to reproduce the appearance of Saturn by fitting an ellipti-
cal " corona " to the planetary globe, meeting it at two places.
They suggested that this corona rotated with the planet once
during its revolution round the sun, on an axis coinciding with
the plane of revolution.

As a result of his work in astronomy and in mechanics
Huygens's reputation was already high. Moreover, his scientific
temper was in accord with that of the best spirits of his age.
" I notice," he wrote to Boulliau (who was an ardent Pythagorean)
on receiving a copy of his treatise on light " that in many places
you dispute the opinions of Aristotle. That is always worth
doing." His opposition to the Aristotelianism of the schools, his
disregard of the Catholic opposition to Copernicanism, and the
steadfastness of his belief in the new mathematical method
brought him the esteem of modern spirits among his contempor-
aries. It was, then, natural that in France, where Colbert was
working to raise the achievements of art and learning above that
of previous ages, Huygens should be considered as one of the
more brilliant among notable foreigners who might be invited to
take up a residence in Paris.

Just as we know little about Newton which is not a descrip-
tion of his mental quality, so it is with Huygens. His corre-
spondence speaks a mind of great intellectual power and clarity.

THE LIFE OF CHRISTIAN HUYGENS 53

and the singular absence of violence and prejudice in his com-
ments on men and things is but a necessary concomitant of this
mentality. Nevertheless, he was a very human creature and one
can sense that the parental authority at times aroused irritation
just as at other times the elderly Constantin's desire to show off
his son caused amusement. The trouble was that old Huygens's
attitude to his sons did not change as they grew up into men, and
when he was forced to treat them no longer as children he
regarded them as young diplomats who might conveniently do
him services in different parts of Europe. Diplomacy was, how-
ever, not much in Christian's line, and while he affected French
elegance and a seriousness of bearing, at the same time he was
impatient with those who were tedious and self-important.

IX

The vicissitudes which both the French and English societies
experienced before receiving official support, and even after, were
such that they might well have died in infancy. Meetings at
Montmor's were discontinued in 1661, but in 1662 the society
held meetings at the house of the Marquis de Sourdis and when
Huygens made another short visit to Paris in 1663 the society had
regained much of its former activity. After the foundation by
charter of the Royal Society in 1662 it was inevitable that in
Paris, where meetings had been held at Mersenne's and else-
where as early as 1650, the idea of a similar institution should be
discussed. Sorbiere, ignoring or ignorant of the early history of
the Royal Society, considered that the early Paris societies had
in fact led the way. But however this may be, he and Huygens
seem to have been on a semi-official errand when they came to
London in 1663 to study the organization of the new Royal
Society. Writing to Boyle of Huygens's introduction to the
Royal Society, Oldenburg said " we had no ordinary meeting;
there were no less than foure strangers, two French and two Dutch
gentlemen: ye French were, Monsieur de Sorbiere and Monsieur
Monconis; ye Dutch, both the Zulichems, 1 Father and Son, all
foure inquisitive after you." Huygens evinced some surprise that
no particular qualifications appeared to be necessary for election
to the Royal Society at this time. Christian accompanied his

* Christian Huygens held the title of seigneur de Zulichem (in the province
of Gueldre) up to the death of his father. He then inherited the title of
seigneur de Zeelhem.

54 THE LIFE OF CHRISTIAN HUYGENS

father on a diplomatic mission on this occasion and was still in
London when the news of his award from Louis XIV was made
public. This necessitated a return to Paris.

During this short stay in England, however, he had occasion
to see further into the character and customs of his hosts.
Through his father's connections in this country he dined a good
deal with the great, and met many personalities outside the
scientific circle at Gresham College. Huygens and his brother
Constantin both dabbled in art, and perhaps his chief interest on
this occasion lay in his visits to the studio of Sir Peter Lely from
whom he obtained a recipe for making pastels.

Soon after his return James Gregory arrived from England
with some correspondence from Moray. Moray wrote of this
young man that he had " a present to make you of a book of
which he is the author, which he calls Optica Promota . . . " He
suggested that Huygens should give his opinion of the work and
its author but Huygens left no record that he did this. The work
was interesting in that it contained a description of a reflecting
telescope some eight years before Newton's invention.

After wintering in Paris, Huygens returned to the Hague
(1664) bent on the pursuit of more fundamental researches than
could be carried out in Paris. Not until 1666, when he became an
official member of the newly formed Academic Royale des
Sciences, did he return. Although, therefore, Huygens's work for
the new academy was very important and the prestige he con-
ferred on it was especially advantageous, it fell to others to com-
plete the details of the organization. A wealthy amateur named
Thevenot gave hospitality to the society at this time and did a
good deal of the preliminary organization. It was he who, doubt-
less with Colbert's knowledge, approached Huygens in
November 1664 with a suggestion that he should become a
member of the reconstituted society. As will be seen, the offer
finally carried with it an official position in Paris with facilities
for scientific work.

Soon after his return to Holland, Huygens set about obtain-
ing patents protecting his design of a pendulum clock for use at
sea for determining longitudes. The news of this move not un-
naturally aroused a good deal of excitement. The commercial
value of a reliable method of finding longitudes at sea would be
enormous and several others were after the prize. The members
of the Royal Society, who knew of previous trials with marine

THE LIFE OF CHRISTIAN HUYGENS 55

clocks, were frankly sceptical about the use of a pendulum clock.
As a result of full discussion, the society, with the national
interest in view, resolved to investigate other methods. Of these
some sort of spring-regulated clock appeared to be the most
promising. It is not surprising, therefore, that Hooke, a most
fertile experimenter, should have taken up the question of the
isochronism of the oscillations of a loaded spring. In August
1665 Huygens heard of Hooke's successful experiments and his
confidence that a spring-regulated clock would be the solution of
the problem.

Huygens returned Hooke's scepticism. So long ago as 1660, he
remarked, the Due de Roannes had tried the idea but without
success. Temperature changes, he considered, would have a
serious effect on the going of such a clock and sufficient accuracy
would be impossible. Hooke, he concluded, spoke too confidently
about this " as also of many other things ". Nevertheless,
Huygens tried a spring-regulated clock in November 1665, but
was hindered through the great delicacy of workmanship
required. Brouncker, in England, found that Hooke's spring
driven spring-regulated clock was not so accurate as a pendulum
clock. The plague interrupted scientific work in London and
Hooke's Potentia Restitutiva, on the properties of springs, did
not appear until 1678. Huygens had to leave the Hague and
retire into the country for a time.

There, at Voorburg, he returned to his work on the com-
pound pendulum, in particular the problem of determining the
centre of oscillation. Lacking a general method, he proceeded
to study the problem inductively, starting with several simple
examples. It was not long before, discarding the erroneous work
of Descartes, he arrived at some " quite pleasant propositions ".
The work aroused great interest in England and its technical
nature will be explained later.

It should be mentioned that by this time (1665) there were two
scientific journals of repute for the publication of new work. The
publication of the Royal Society, Philosophical Transactions, was
begun by the secretary, Oldenburg, on his own initiative in
March 1665; in Paris the Journal des Savants was started, also as
a private venture, by de Sallo in January of the same year. De
Sallo's privilege was withdrawn after about a year because of his
denunciation at Rome, but the Abb6 Gallois restarted the journal
in January 1666. Neither the Journal nor the Transactions had

56 THE LIFE OF CHRISTIAN HUYCENS

the form of modern scientific periodicals: little original work
was published and they were more of the nature of reports. The
second number of the Transactions bore an account of some
observations by Guiseppe Campani on Saturn's ring. These were
of interest since Campani claimed to have distinguished the
shadow thrown on the planet by its ring, the remarkable thing
really being that his telescope was sufficiently good for such
detail to be seen. It was said that Campani's lenses were ground
and polished on a machine, but attempts so far made in this
direction had been discouraging. Hooke published an account
of a machine but it does not appear to have been well tested and
he was castigated for publishing an account " upon a meer
theory".

Cassini, using a telescope made by Campani, observed a per-
manent mark upon the surface of Jupiter and from its return was
able to give the period of revolution. This, Huygens affirmed,
was " assuredly a very fine discovery ". He himself succeeded in
observing the shadow of one of the satellites of Jupiter on the
surface of the planet as predicted by Cassini. He also spent some
time studying a comet which made its appearance at the end of
1664. As will be explained later, the paths of the comets, so far
as these were known, were proving a great difficulty for
Descartes's cosmology. Huygens was primarily interested in them
as they concerned the Copernican theory. He was at first sceptical
about the idea that they recur at long intervals of time. It is
interesting that Horrox's defence of the Copernican theory,
written about 1635 and resuscitated by members of the Royal
Society, came to Huygens's notice through his correspondence
with Moray. Horrox, although he died at the age of twenty-two,
is generally agreed to have made his mark as an astronomer of a
very high order.

The largest telescopes used at this time were of the type now
known as Huygens's aerial telescopes, but it is not clear that he
originated the idea or wished to claim it as his own. Auzout
used an aerial telescope and devised his own method of aligning
the lenses; in England the suggestion was widely attributed to
Wren. An invention of greater importance and one to which
Huygens made an interesting contribution was the micrometer
eyepiece. It began to be realized that telescopes could be used
for the determination of small quantities which were completely
beyond the scope of ordinary instruments used up to that time

PLATE III

Huygens's Clock as the Centre Feature of a design showing
Scientific Apparatus of 1671

THE LIFE OF CHRISTIAN HUYCENS 57

for quantitative work. The measurement of small angular
separations, for example, required the use of a very large
quadrant, but these large instruments became distorted under
their own weight. Gascoigne first hit on the idea of using two
fine hairs close together and situated in the focal plane of the
objective as a means of converting the telescope to quantitative
measurements. Auzout and Huygens did some measurements
of planetary diameters in 1664 an d 1665, but Huygens's micro-
meter was a thin plate of metal in the form of a trapezium. It
was inserted between the two lenses of his eyepiece where the real
image was formed. The plate could be moved until the disc of
the planet was just obscured. In this way, as early as December
1659, h e obtained a good result for the diameter of Mars. Shortly
after Huygens went to Paris in 1666 a micrometer consisting of
moveable hairs was used. The modern form of micrometer was
invented by Auzout and Picard. Curiously enough it was Picard
who saw the value of the pendulum clock in astronomy rather
than Huygens. Delambre remarks that Huygens " started the
great revolution " in practical astronomy by his invention of the
pendulum clock but it was Picard who did most to introduce
regular time observations at the Paris observatory. Using
Huygens's pendulum clock he used the times of meridian transit
of stars to determine their differences in right ascension.

Huygens was not, in fact, a regular observer. His contribu-
tion to astronomy lay rather through his work on optics, which
had throughout a practical bias: the invention of his eyepiece
and the study of conditions under which spherical aberrations
may be reduced. In this period just preceding his departure for
Paris, Huygens became deeply interested in two works sent over
from England : Hooke's Micrographia (1665) aad Boyle's Experi-
ments and Considerations touching Colours (1664). Hooke,
indeed, was at his best in descriptive and experimental work in
which mathematics was not required. The hypotheses which he
and Boyle advanced regarding the nature of light and the cause
of colours were extremely stimulating and aroused Huygens to
the desire to carry out experiments on the subject. From these
days some of his important work in physical optics may be dated.
He was convinced that before the phenomena of colour could be
explained it would be essential to understand the mechanism of
refraction. This, he considered, Hooke and Boyle had omitted to
study sufficiently. His own note-books show that he calculated

58 THE LIFE OF CHRISTIAN HUYGENS

the order of thickness of the air film involved in the production
of colours by interference in the so-called Newton's rings experi-
ment (November 1665). Boyle, while admitting that he knew of
this experiment, wisely declined to be drawn into a discussion of
its explanation.

In his views on the nature of light Huygens always showed a
greater dependence on Descartes than in the rest of his work.
This bias may explain his first scornful reception of Fermat's
least-time principle, for Fermat was, of course, the great critic
of Descartes 's work in optics. His principle that a ray of light
follows that path for which the time of transmission is less than
for any alternative path had also an Aristotelian flavour, or so it
seemed to Huygens. He declared he found no satisfaction in the
idea and considered it was a " pitiable axiom ". Nevertheless, he
repeated Fermat's calculation of indices on this " obviously pre-
carious " principle and, while retaining doubts as to its validity,
began to be convinced more and more that the refractive index
of a medium is in fact given by the ratio of the velocities of light
in air and in the medium. It was necessary to suppose, with
Fermat, that light has a finite velocity, whereas Descartes staked
his scientific reputation, as he said, on the belief that its velocity
is infinite. Roemer's famous calculations of 1676-7 were thus
extremely important, for they showed that Fermat and Huygens
were correct.

In the meantime, as has been mentioned, the men of science
in Paris had not found it easy to get the project of a permanent
academy of science properly launched. The intimations Huygens
received of a position in such an academy were not for a time
followed by any concrete offer. Nevertheless, his name was kept
in front of Colbert. Moray wrote to Oldenburg in 1665 that
"Colbert intends to sett up a Society lyke ours and make
Huygens Director of the designe," but during this year Huygens
began to feel far from confident about the statements which
reached him from Chapelain. He bombarded Carcavy with
anxious letters and his feelings had to be assuaged with a variety
of excuses. No doubt official delays occurred and accommoda-
tion had to be found. Huygens was, however, more concerned
over the amount of his salary, clearly through anxiety to live in

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THE LIFE OF CHRISTIAN HUYGENS 59

the style to which his upbringing and habits had accustomed
him. When he arrived in Paris in 1666, it was to find that no
plans for the new society had been drawn up: The official found-
ing of the Academic Royale des Sciences, on June ist, meant at
first nothing by way of financial aid. It was simply agreed that
Auzout, Roberval, CarcAvy, Frenicle, Picard, Buot and Huygens
should be the nucleus. But, from a letter l written by Montmor
to Alessandro Segni, the secretary of the Accademia del Crusca,
it looks as if it was from the first intended that Huygens should
have a prominent place. Then, too, writing to Hevelius in 1667,
Boulliau said: "Chief of all is the renowned Christian
Huygens . . . Next are Roberval . . . Auzout ..."

Huygens became a close friend of the Colberts and served on
occasion as the minister's scientific adviser. Meetings were held
at first in Colbert's library and the first co-operative undertaking,
an observation of a lunar eclipse, June 16, 1666, was planned to
take place at his house. Unfortunately, cloudy weather made
observations impossible. But two weeks later the same group,
Huygens, Carcavy, Roberval, Auzout, Frenicle and Buot met to
observe a solar eclipse. Once more visibility was poor and results
were disappointing. Micrometer measurements giving the rela-
tive diameters of the sun, moon and planets were, however, made.

The appearance of the members at these early gatherings
of the Academic Royale has been excellently recorded in the
work of the skilful engraver Lfe Clerc. One of this artist's
pictures, for example, shows an informal meeting of members
and may well represent an hour spent in desultory discussion
before the giving of an address. It has been suggested that the
figure holding a lens and standing in the window is that of
Huygens. When we look at the plate showing a meeting
attended by Louis XIV (facing p. 60) it does not appear that
Huygens was included. This is the opinion of E. C. Watson,*
who points out that Huygens was away from Paris through
illness early in 1671.

In August 1666 Huygens took over apartments at the
Bibliothique du Roi after the headquarters of the Academic had
been transferred there. On December M the society gathered
officially for the first time to hear from Carcavy the decision of
the king to protect the new institution.

i A. J. George. Annals of Science, III, 37*.
* E. C. Watson, Osiris, VII, 556.

60 THE LIFE OF CHRISTIAN HUYGENS

Adherents of Descartes's philosophy, men, that is, who gave
out their belief in vortices of subtle matter and who did
not accept atoms or the existence of a vacuum, were not con-
spicuous in the make-up of the Academic. Roberval was a noted
critic of Descartes; later the Academic included Mariotte, who
also was dubious about Cartesian theories. Frenicle resembled
Mariotte in being prepared to accept resemblances between facts
without feeling obliged to attribute them prematurely to a single
cause. Huygens alone referred to the doctrines of Descartes's
Principia when called on for explanations of such phenomena as
gravity, and he and Charles Perrault for a time exerted a slight
influence in favour of Descartes. In time, it is clear, Huygens
became distinctly aware of the failure of such an experimen-
talist as Rohault to maintain his Cartesian explanations without
disingenuousness,and his work shows a progressive decline in his
adherence to the great " system ". Yet he long remained in two
minds and it only required some ignorant criticism of the great
philosopher to rouse him to his defence. It is surprising too, to
see how closely the form of some of his work (for example, that
on impact) resembled that of Descartes 1 .

Unfortunately this period of Huygens's work in Paris was
twice interrupted by serious illness, necessitating a return to his
native air. One gains the impression that his position was thereby
weakened in some degree, for his absences were prolonged. When
he left for yet a third time to regain his health it was never to
return. His protector Colbert died soon after and profound
changes in the p critical situation militated against his regaining
a position which all along had aroused a certain envy. Huygens's
last years were consequently spent in comparative retirement.
Mach says that Huygens " shares with Galileo a noble, unsur-
passable and complete uprightness " and this is a true estimate.
The manner in which his years at Paris came to an end can only
be deplored. Huygens's scientific work throughout illustrates a
readiness to make his personal reputation always subservient to
larger interests. Nil actum reputans, si quid superesset agendum
was, according to the historian PoggendorfF, his adopted device.

The presence of Huygens in Paris throughout the onslaughts

of the French armies against the Dutch Republic is the fact

which historians find hardest to explain. To a large extent wars

reflect nothing of individual feeling towards members of another

l Cf. Mouy. Le Dtveloppement de la Physique Cartdsienne 1646-1712 (p. 197).

PLATE V

Louis XIV at a Meeting of the Academic

THE LIFE OF CHRISTIAN HUYGENS 6l

nation and in those days the instruments of propaganda
necessary for whipping up appropriate hatreds did not exist.
Nevertheless, in the shifting scene of the wars of Holland, now
against England with France as ally, now against France, then
against both, and later with England as an ally, the opposition to
Louis XIV really remained the one permanent feature. Louis
was bent on destroying the Dutch Republic and, with the
treacherous help of Charles II, it looked in 1672 as if he would
succeed. Huygens could have secured a position of eminence
under the Prince of Orange at this time but he had a deep
repugnance for political activity and remained in Paris, suspected
by some but protected throughout by the minister Colbert. It
is not surprising, therefore, that he came in for some criticism by
his fellow-countrymen. This criticism was brought to a head in
1673 by the eulogistic dedication of his great Horologium
Oscillatorium to Louis XIV. The explanation of these facts
seems to be that, once having yielded to the cordial friendship of
his associates at the Academic Royale and having decided to
endure the war, he had to pursue a difficult and always depress-
ing course. The dedication may be regarded as a piece of
political wisdom, as justifying his continued patronage, in fact.
As a friend of the Dutch ambassador van Beunigen, who during
the short war of 1667-8 was suspected of a plot against Louis, it
would have been easy for him to come under suspicion as a spy.
For Huygens was by no means remote from the world of affairs.
He was very well known at court and had many influential
friends. After the rise of Louvois and the death of Colbert none
of these things mattered; the feelings of Louis towards the house
of Orange can scarcely be said to have improved after 1678.

But in 1666 Huygens was indisputably the one who chiefly
guided the affairs of the Academic Royale des Sciences. Profiting
from his knowledge of the Royal Society, Huygens emphasized
in Paris the importance of Bacon's teaching. " Experiment and
observation/' he wrote, " provide the only way of arriving at the
knowledge of the causes of all that one sees in Nature." This
attitude is the more striking when one reflects that Descartes
had so long been his model. The point here is that while both
Bacon and Descartes distrusted formal logic, Descartes scorned
empiricism while Bacon apprehended its power. It is not clear
that Huygens realized the shortcomings of Bacon's " method ".
The great omission in Bacon's scheme of research was the recog-

62 THE LIFE OF CHRISTIAN HUYCENS

nition that measurements provide the key to the understanding
of phenomena. Bacon ranged himself with Aristotle in saying
classify when he should have said measure.

The attention paid to Chemistry should, in Huygens's view, be
restricted to essential problems. He appears to have recognized
that the old alchemy was decadent and that the beginnings of
a true science lay in the work of Boyle and others. Problems
such as that of combustion were obviously worth the closest
study. Huygens was interested in Hooke's experiments at the
Royal Society which, he held, agreed with the " bizarre hypo-
thesis " of an " aerial saltpetre ". This hypothesis of an active
constituent in the air he considered was " not ill conceived " but
he tended towards Moray's empiricism. " We others/' wrote the
latter, " look for the truth of existence and the nature of things
as belongs to the true philosophy ".

There can be no doubt of the influence of the London group
on Huygens 's views of the functions of the new Academy. This
influence abroad was recognized by the English themselves, who
were fully conscious of the unique importance of their work.
" I hope our Society will in time ferment all Europe at least,"
wrote Oldenburg to Boyle. " Let envy snarl," he wrote, when
the new societies excited opposition, " it cannot stop the wheels
of active philosophy in no part of the known world."

The Academic at first made astronomy its special study, en-
couraged, no doubt, by the occurrence of a partial eclipse of the
sun in 1666. Huygens noted with dissatisfaction the paucity of
astronomical observations in earlier years and this was to be
remedied. New observatories were in the course of construction
at Greenwich and Paris; Hevelius at Danzig had for some years
applied himself to completing Tycho Brahe's observations and
had, in 1661, made with Boulliau careful observations of a solar
eclipse. In 1666 more was expected from eclipse observations,
namely, to rectify the motion of the earth and the moon and to
determine differences of meridian on the earth. In the course of
the work, several telescopes were compared and micrometers were
used for obtaining the relative diameters of the moon and sun.

In continuing his work on lenses at Paris, Huygens was hin-
dered by the poor quality of the French glass, which was inferior
to the Venetian. The material showed veins or striae and tended
to extrude salts on cooling. Lens-making brought Huygens into
contact with the work of Spinoza, who had then a greater reputa-

THE LIFE OF CHRISTIAN HUYGENS 63

tion as a lens-grinder than as a philosopher. Lenses were ground
by hand in a hollow form or mould in which abrasives of increas-
ing fineness were successively used. It was the impossibility of
grinding any but small lenses in this way that put a limit to the
power of telescopes. The appearance of colours in the image was
considered by Huygens to be connected with the inclination of
the lens surfaces. The error in this was recognized by Newton
some years later but in the meantime a great deal of work was
expended in the attempt to make lenses of other than spherical
curvature.

Astronomy in Paris gained very greatly by the arrival of
Cassini in 1669. His first observations were made at the new
observatory in 1671. Here he continued his striking work on the
rotation of certain of the planets. Huygens had observed the
rotation of Mars in 1659 ^ ut true to his device, had not con-
sidered the results sufficiently good for publication. Cassini was
rewarded by the discovery of four satellites of Saturn and the
division in Saturn's ring which is now known by his name.

Huygens was not a competitor with Cassini for the honours of
new astronomical discoveries. After 1666 his interests lay more
in the direction of terrestrial mechanics and, as the sequel shows,
this preference was sound. " I am now starting experiments on
circular motion," he told his brother in 1667. A few years later,
when Richer's expedition to Cayenne returned to Paris it brought
back interesting evidence which bore on the question of the
earth's gravity, but the effect of circular motion of a medium
was the question which at this time interested Huygens, for it
was through this that he hoped for an explanation. In taking up
the effects of rotation, Huygens was, from one standpoint, return-
ing to work which he had put away ten years before. By 1659,

v 2
it is thought, he had arrived at the expression for the

acceleration towards the centre in the case of a body describing a
circular path. In 1669, however, he chose to address the Academic
Royale not on this but on an elaborate theory employing the
vortex of subtle matter as the cause of gravity. Any easy con-
victions we may have that Huygens had by this time rid
himself of Cartesian influences must be profoundly shaken by
a perusal of this discussion. The opposition with which his
theory was greeted by Mariotte and Roberval on this occasion
may have been highly beneficial, for the criticisms they made

64 THE LIFE OF CHRISTIAN HUYGENS

were entirely justified. It may be mentioned in passing that
Huygens at this time believed that circular motion is a funda-
mental form. Uniform rectilinear motion, he saw, had no effects
on events which normally occur in an apparently stationary
environment. Circular motion, howfever, introduced new effects.
It was only after the appearance of Newton's Principia that
Huygens retracted this statement of the absolute nature of
circular motion. He then, more consistently, took a firm stand on
the relative nature of all motion and against the idea of any
absolute space.

Mariotte was a French priest who joined the Academic
Royale in the year of its foundation and thereafter played an
important part. He must be reckoned among the lesser lights
who at this time were attempting to make the important next
step beyond the mechanics of Galileo. His TraiU de la percussion
ou choc des corps (1677) shows that he and Huygens were work-
ing on similar problems. When Oldenburg approached Huygens
in 1668 with a request that he should contribute to the Royal
Society some work on mechanics he replied by sending some
work on impact. This, afterwards published in the posthumous
Tractatus de Motu Corporum ex Percussions (1703), is really a
study of various applications of the law of conservation of
momentum. There can be no doubt that Newton profited from
the work on impact which was carried out by Huygens, Mariotte,
Wallis and Wren. The formulation of his relation between rate
of change of momentum and external impressed force completed
in a magnificent way this contemporary work. In regard to centri-
fugal force Huygens forestalled Newton by many years. " What
Mr. Huygens has published since about centrifbgal force I sup
pose he had before me," wrote Newton with some chagrin.

The immediate result of the correspondence with Oldenburg
was that Huygens learnt that Wren and Wallis had both com-
municated papers on the subject of impact and momentum and
at practically the same time as his own. More instances were to
come in which Huygens felt himself to have been unfairly fore-
stalled in publication and in some cases he gave vent to severe
criticisms which were by no means justified. Oldenburg showed
great fairness and removed much oif this feeling of resentment,
but the outbreak of some acrimonious correspondence over some
mathematical work by James Gregory shows that Huygens had
become unduly nervous for his reputation. When Mercator put

THE LIFE OF CHRISTIAN HUYGENS 65

forward a method of determining longitude by means of a pen-
dulum clock he roundly condemned his intrusion. It was for-
tunate that when Barrow's Lectiones Opticae came out in 1669 it
was evident that the work did not overlap with Huygens's pro-
longed researches in optics. Huygens was surprisingly slow to
learn the consequences of his own attitude towards publication.

It was not customary in those days to isolate a particular
problem and to study it exclusively for a considerable time. The
seventeenth century men of science were nearly all capable of
turning their attention to a wide range of subjects and they fre-
quently were engaged on a variety of topics. Huygens indeed
must be considered one of the most versatile men of the age,
for he excelled Hooke in the quantitive nature of his work while
at the same time he showed as wide a range of activity. Hooke's
Micfogrophia stimulated Huygens at this time to attack the
problems of constructing microscopes, employing the theoretical
advances he had achieved in his work on the telescope. Spinoza
was interested in similar problems. Galileo was described as
having constructed " an occhiale which magnifies ... so that one
sees a fly as large as a hen ". This was a compound microscope.
Hooke improved the instrument as regards its mounting and the
illumination used. Optical improvements were seriously needed.
Huygens's own microscopic observations will be mentioned later;
they belong to the years after his translation of Leeuwenhoek's
work into French in 1677 or 1678.

As has been mentioned, Huygens's health was never robust.
From early youth he was from time to time subject to a certain
kind of debility, later accompanied by severe headaches. The
illness of 1670 brought about his complete prostration in Paris
and he clearly believed himself to be at the point of death. In
these circumstances he concluded that he should bequeath his
more important unpublished work in mechanics to someone
capable of appreciating its importance and he decided to send it
to London in the hands of Francis Vernon, secretary to the Eng-
lish ambassador. This action is sufficiently interesting in view of
Huygens's official position in Paris for Vernon's account to be
given at length. In a letter to Oldenburg he described Huygens's
condition "... I saw the condition hee was in which was none of
the most lively, that his weaknesse & palenesse did sufficiently
declare how great a destruction his sicknesse had wrought in his
health and vigour & that though all was bad, which I saw, yet

66 THE LIFE OF CHRISTIAN HUYCENS

there was something worse which the eye could not perceive nor
sense discover, which was a great dejection in his vital spirits, an
incredible want of sleep, which neither hee, nor those who coun-
celd and assisted him in his sicknesse knew how to remedie &
that hee did not know what the end of these things would bee,
butt his fancy was ready to suggest the worst ..." This mood
Vernon did his best to dispel. He accepted his mission to England
should the worst befall. Then, he wrote, " hee fell into a discourse
concerning the Royal Society in England wich hee said was an
assembly of the Choicest Witts in Christendome & of the finest
Parts: hee said hee chose rather to depositt those little labours
of his which God had blest and those pledges which to him were
dearest of anything in this world, in their hands sooner than in
any else. Sooner then of those into whose Society hee was here
incorporated & from whom hee had received all demonstrations
of a most affectionate civilitie because hee judged the Seat of
Science to bee fixed there & that the members of it did embrace
& promote Philosophy not for interest, not through ambition or
a vanity of excelling others not through fancy or a variable
curiosity, butt out of naturall principles of generosity, inclina-
tion to Learning & a sincere Respect and love for the truth. . . .
Whereas hee said hee did foresee the dissolution .of this academic
because it was mixt with tinctures of Envy because it was sup-
ported upon suppositions of proffitt because it wholly depended
upon the Humour of a Prince & the favour of a minister, either
of wich coming toe relent in their Passions the whole frame &
Project of their assembly cometh to Perdition/'

It is clear that so early as 1670 differences had arisen between
Huygens and certain members of the Paris Academy. This fact
will be of interest later when the circumstances of the rupture
of his official connection are considered.

Huygens's illness lasted in acute form for several weeks during
which great anxiety was felt by his friends in Paris and London.
In June there were signs of recovery and three months later
the convalescent was able to return to the Hague. In October he
resumed correspondence with Oldenburg.

XI

Apart from this winter in Holland, 1670-71, the five years
from 1670 to 1675 were spent by Huygens in Paris. And they were
stirring years in the scientific world. Huygens as chief of the

THE LIFE OF CHRISTIAN HUYGENS 67

Paris Academic was at the centre of things on the Continent,
while he was well informed of what passed in England. In 167 1
Picard's M&ure de la Terre came out, a work of interest from
the technical aspect as well as for a general discussion of current
theories concerning the shape of the earth. Picard, for example,
was well aware in 1671 that the length of a seconds pendulum
was different at London, Lyons and Boulogne, but while he
admitted that the results might be in conformity with the rota-
tion of the earth, he did not think there was sufficient confirma-
tion of the results, as yet, to justify any conclusion. Very probably
he was acquainted with the notion of centrifugal force through
his relationship with Huygens, for the latter had arrived at his
important theorems as early as 1659. I* 1 l fy* news came from
England of Newton's work on the solar spectrum; from Holland
in the same year came interesting mathematical work by Slusius
on the drawing of tangents to curves. 1673 was the year of Huy-
gens's Horologium Oscillatorium, his magnum opus. In 1674
Hooke issued a work giving his views on evidence for the
motion of the earth.

During this period Huygens worked with Denis Papin on
the use of gunpowder as a source of useful energy, and, more
important, with Leibnitz at Mathematics. In 1675 Leibnitz
brought out his calculus differentialis. And to these busy years
belong also the invention of the spiral-spring regulator and
balance wheel which are essential parts of the watch and chrono-
meter.

Yet they were not altogether happy years for Huygens. After
the invasion of the Low Countries, by the armies of Louis XIV
in 1672, he may frequently have asked himself why he had re-
turned to Paris in 1671, and immersed himself in work as the
only outlet for his despair at his situation. Huygens felt keenly
the wrong done to his country and it required much tact and
consideration from his friends in Paris to preserve the calm rela-
tionship in which he had been accustomed to live. Huygens
followed the course of the war with anxiety, but it must be
remembered that at forty-three he was by upbringing and experi-
ence almost as much a citizen of the French capital as he was of
Holland. Paris was indeed the centre of the cultivated world and
the prospects for the man of science who should be cut off from
the activity of one or other of the two flourishing societies would
be poor indeed.

68 THE LIFE OF CHRISTIAN HUYGENS

Huygens seems to have worked even when quite ill; certainly
he was struggling to regain his old activity in October 1670 when
he received some interesting work on mechanics from Wallis.
Yet he agreed that his recovery (" for which I thank God ") was
too recent for him to do other than attempt very little.

We shall see that Huygens's work in theoretical optics, for
example, rivals that in mechanics, but in 1670 he considered the
latter subject more important. Though overlooked nowadays,
the ideas he contributed to mechanics were as fundamental as
his contributions to physical optics. His Horologium Oscilla-
torium was practically completed and had grown from a treatise
on the construction and regulation of the clock to a work on
centres of oscillation, the tautochrone, the theory of evolutes
and centrifugal force. As an examination of this treatise will
show, it contains a great deal which was made more explicit in
Newton's Principia (1687) although discovered by Huygens
independently. The law known as Newton's First Law was known
and used by Galileo and Huygens; Huygens, in addition, must
have employed the Second Law in arriving at his propositions
on centrifugal force as early as 1659. He also saw the necessity
of distinguishing between mass and weight at about the same
time (see p. 119). The greater merit of Newton's work, in
fact, was that he gave a clearer presentation of these ideas and
made them more useful by means of simple mathematical
relations.

When the Horologium Oscillatorium came out in 1673, after
Huygens's return to Paris, it showed the extent to which his
thought had developed. The work was singularly free from Car-
tesian influences. Huygens himself hoped that it would be in
direct line with the great work of Galileo and his hopes were not
disappointed. Newton wrote to Oldenburg of his " great satis-
faction " with the work and said he found it " full of very subtile
and usefull speculations very worthy of ye Author ". Newton
especially admired Huygens's mathematical style and con-
sidered him the " most elegant writer of modern times ". This
remark starts some interesting reflections. Newton regretted that
he had not applied himself to geometry before proceeding to
algebraic analysis. It was Huygens's predominantly geometrical
methods, employed in the Horologium Oscillatorium, which
aroused his admiration. At this time Newton was well advanced
in his work on fluxions and, as w-e know, Leibnitz took up similar

THE LIFE OF CHRISTIAN HUYGENS 69

problems after 1672. The central idea of the differential calculus
owes a great deal to the study of motion, for this study intro-
duced the notion of a continuously varying quantity. Huygens 's
work in this connection was of the greatest importance for, as
Leibnitz admitted, it was Huygens who had dispelled the rnys
tery attaching to the study of motion.

The two mathematicians, Huygens and Leibnitz, met in
Paris in 1672 and Leibnitz became a regular visitor at the
Bibliotheque du Roi. Under Huygens's guidance Leibnitz's ideas
developed rapidly, for up to this date, as he himself admitted, he
had been only an amateur in such studies. In 1674 Huygens was
able to present to the Academic Royale Leibnitz's first paper on
the differential calculus. Whether Huygens gave Leibnitz an
inkling of Newton's work on fluxions will always remain an in-
teresting speculation. Newton's own ideas date from about 1665
or 1666 and there is no doubt that after 1669 these were well
known to his friends in England. Wallis especially must have
known about them. Huygens himself was not happy in the use
of analytical methods. He was, in Newton's words, " the most
just imitator of the ancients " and it is a striking fact that the
classical geometrical method was used by Newton himself in
writing the Principia. This fact, which has always troubled
historians in some degree, must be explained by the prestige
of Huygens at this time and the fact that proofs by the newer
methods were not everywhere accepted. The ideas of both the
Principia and the Horolofnum Oscillatorium were later cast in
analytical form by the mathematicians of the eighteenth century.

The Horologium Oscillatorium made a great impression on
contemporary men of science. The propositions on centrifugal
force, given at the end, were of course important in the develop-
ment of planetary theory, and the conical pendulum interested
those who, like Hooke, concerned themselves with the problem
of time measurement. It is certain that Huygens employed the
conical pendulum in clocks in 1659 and again in 1667, when he
had more fully investigated the laws of motion involved. Con-
troverting Hooke's claims to the invention of such a clock, he
pointed out, what Hooke certainly did not know, that the conical
pendulum should be so designed that all revolutions of the bob
describe horizontal circles in the surface of a paraboloid of revolu-
tion with the axis vertical. Only then would all revolutions be
isochronous.

70 THE LIFE OF CHRISTIAN HUYGENS

Huygens showed considerable dislike for quarrels such as the
one of 1674 in which he became involved with Hooke and others.
In 1675 he had yet another such experience. This was over his
invention of the first successful spring regulator for the clock.
Huygens's design of a spiral spring combined with a balance
wheel is the one which is still used in watches and its distinction
lies in the fact that the centre of gravity of the oscillating part
bears a fixed relation to the stationary parts. This meant that the
influence of gravity was eliminated. A quarrel with the clock-
maker Thuret and the Abbe de Hautefeuille was settled by the
Academic Royale on these grounds in Huygens's favour, for
Hautefeuille used a straight spring and not a spiral. Hooke was,
however, a more tedious antagonist in connection with the same
invention and he was made more bitter by the support given by
some members of the Royal Society for Huygens's priority.
"Zulichem's spring not worth a farthing," he wrote in his
diary. When Huygens, for a quiet life, gave Oldenburg the rights
to the English patent, he drew from Hooke a full and free ex-
pression of his long dislike of the secretary. Oldenburg, he
averred, was Huygens's spy. " Saw the Lying Dog Oldenburg's
Transactions," he noted eight months later, " Resolved to quit all
employment and to seek my health." Oldenburg, he said angrily,
was a "trafficker in intelligence". He would hear nothing favour-
able about Huygens's marine clocks. " Seamen knew their way
already to any Port," he somewhat childishly stated. Altogether
the complaints of Hooke appeared in print over a considerable
period tut they hold little interest now.

A more famous result of Huygens's correspondence with the
English scientists during his years in Paris was that the French
became aware in 1672 for the first time of the work of Newton.
On January 1 1 of that year Oldenburg wrote to Huygens of the
" invention of a new sort of telescope by Monsieur Isaac Newton,
Professor of Mathematics at Cambridge ". His next letter gave a
full description and this was utilized by Huygens for an account
published in the Journal des Savants of the following month. It
should be mentioned that Gregory's design for a reflecting tele-
scope was never put into practice and the new invention was
based on his ideas. Huygens found it *' beautiful and ingenious "
and he thanked Oldenburg for the news of " the marvellous tele-
scope of Monsieur Newton ". The problem of making the con-
cave mirror, though difficult, did not appear to be insuperable.

THE LIFE OF CHRISTIAN HUYGENS 71

He supposed that Newton had come to despair, as he had him-
self, of overcoming spherical aberration but did not refer to the
fact that it was this consideration which had led Gregory to his
idea nine years before. Up to this date Huygens had not heard
of Newton's experiments on the composition of white light and
certainly underestimated the defect of chromaticism.

It is consequently surprising, after this initial enthusiasm, to
find that Huygens soon abandoned the reflecting telescope. The
trials which he himself immediately carried out proved dis-
appointing owing to the imperfect polish given to the mirror.
Newton's first telescope was hardly more than a model and when
the construction of larger telescopes was attempted in England,
the most expert glass worker in London, Cox, found the task of
grinding the mirrors too difficult. Reflecting telescopes of a large
and useful size were in fact not made for more than fifty years.
Huygens found that metal jnirrors were unsuitable since the
polish was unequal to that given to glass and it was not perman-
ent. He found himself compelled to return to the refracting tele-
scope but realizing that a new difficulty beside that of spherical
aberration now required to be overcome.

For in March 1672 Oldenburg sent Huygens a copy of the
Philosophical Transactions in which, he said, Huygens would
find " a new theory of Monsieur Newton (the inventor of the
cata-dioptric telescope) touching light and colours : where he
maintains that light is not uniform but a mixture of rays of
different refrangibility, as you will see fully in the same dis-
course. . . " This copy of the Transactions contained, of course,
an account of Newton's famous experiment on the spectrum.
The Royal Society, on receipt of this, Newton's first published
scientific paper, sent it to Huygens as the one whose opinions
would carry the most weight. Huygens's reply was that the con-
clusions drawn and the theory put forward seemed "very
ingenious ". But, he went on, " it must be seen if it is compatible
with all the experiments ". Three months later he wrote that he
considered the compound nature of white light had been proved
by Newton's experimcntum crucis, in which it was shown that
the separate coloured rays emerging from the prism could not
be further decompounded. Huygens went on, however, to make
observations which disappoint the modern reader almost as much
as they disappointed the young Newton. He questioned if it
would not suffice to base an explanation " on the nature of move-

72 THE LIFE OF CHUISTIAN HUYGENS

ment " for the two colours yellow and blue only. Until the
essential difference of these two colours was understood " he
[Newton] will not have taught us what the nature and difference
of colours consists of, but only this accident (which assuredly is
very considerable) of their different refrangibility ". Failing to see
the distinction between an impression of colour and the different
rays of the spectrum, Huygens suggested that Newton would
find that yellow and blue would be sufficient to produce white
light. The other colours he regarded as " degrees of yellow and
blue more or less deep ".

These criticisms were an easy prey to Newton who, far from
rushing prematurely into publication, had kept the work by him
for at least five years. Oldenburg warned Huygens that Newton,
then thirty, was not a man who spoke lightly about anything
he advanced. Newton flatly denied that all colours could be " de-
rived out of the Yellow and Blew . . . none of all those colours
which I defined to be Original " could be so obtained. " Nor is
it easier,." he insisted, " to frame an Hypothesis by assuming only
two Original colours rather than an indefinit variety; unless it
be easier to suppose, that there are but two figures, sizes and de-
grees of velocity or force of the ^Ethereal corpuscles or pulses,
rather than indefinit variety; which certainly would be a harsh
supposition." It would be indeed, he remarked, " a very puzzling
phenomenon ", "... But to examine how Colors may be ex-
plained hy pot helically is besides my purpose. I never intended to
shew, wherein consists the Nature and Difference of colors, but
only to shew, that de facto they are Original and Immutable
qualities of the Rays which exhibit them; and to leave it to
others to explicate by Mechanical Hypotheses the Nature and
Difference of those qualities: which I take to be no difficult
matter ". It was unimportant if two colours in the spectrum
could be combined to give an appearance of white. Such light
was different in its physical nature from ordinary white light
and could not be resolved by the prism into more than the
two components. Clearly Newton was deeply disappointed, at
the outset of his career, to receive so little appreciation of the
true nature of this work from one so eminent as Huygens. He
wished, he said, in future " to be no further solicitous about
matters of Philosophy ". His rather summary answers to Huy-
gens's remarks disposed at the same time of certain criticisms
put forward by Hooke. For once Huygens and Hooke were in

THE LIFE OF CHRISTIAN HUYGENS 73

alliance, both opposed to what was essentially a new attitude
to scientific problems; both, moreover, found the new
facts difficult to reconcile with their respective wave or pulse
theories of light. This first occasion for the comparison of Huy-
gens and Newton raises, it will be seen, a question on which these
two men of science consistently differed. This was the place of
hypothesis in scientific method, a subject for which a later chap-
ter must be reserved.

It is unfortunate that the two greatest scientists of this period
did not achieve harmony in their attitude to problems of com-
mon interest. For their divergence was not particularly fruitful
although it extended from optics into the realm of mechanics.
Huygens greatly admired the Principia after its appearance in
1687 and he met Newton in 1689. By r his time, however, the dif-
ference of outlook had become too ingrained and Huygens at
sixty had become less amenable to new persuasions. And yet
this difference is certainly not so great as some writers have sug-
gested. One biographer of Newton, Louis Trenchard More, con-
siders that men like Hooke and Huygens relied on an inward
sentiment of knowledge and in opposing Newton " were merely
opposing theory by hypothesis ". Whatever the significance of
this distinction, this is a question which can be dealt with only
after a careful survey of Huygens's work as a whole. It will be
seen that it is a profound mistake to treat Huygens as merely
another Cartesian, for all his life he vacillated between the Car-
tesian view that the objects of scientific calculation are products
of thought and the materialism which regards them as external
realities. Not only is there much to say in Huygens's favour with
regard to the status of scientific concepts, but in methodology
also Huygens perceived as well as Newton the end of scientific
investigation. " I do not believe we know anything with com-
plete certainty," he wrote to Perrault, " but everything probably
and to different degrees of probability ... as 100,000 to i as in
geometrical demonstrations/* The latter he considered were in
a category by themselves. " In the matter of Physics there are
no certain demonstrations and one can only know causes through
the effects in making suppositions founded on experiments or
known phenomena and trying afterwards if other effects agree
with these same suppositions." These remarks should make clear
the difference between Huygens and Descartes. For Descartes
the intuitional method, to which More refers, did undoubtedly

74 THE LIFE OF CHRISTIAN HUYGENS

take precedence over the experimental. For Huygens, a life-long
experimentalist, there was no way to final certainty.

In the period we are considering Huygens continued his
studies in optics but it is difficult to estimate his progress. This
work began as early as 1652. By 1653 he had completed a first
Tractatus de refractione et telescopiis of 108 pages. This was sub-
sequently amplified but remained unpublished. As Huygens
found, Cavalieri had independently obtained some of his results
and published them in the Exercitationes Geometrical Sex. The
only other previous writers of originality were Alhazen (nth
century) and Kepler, whose Paralipomena was published in 1604
and Dioptrice in 161 i . In England the most important work was
done by Barrow and by Halley. The latter drew attention to the
advantages of algebraic formulae; up to this time the relations
used in optics were expressed in the most cumbersome geometri-
cal form. Many of the problems to which Alhazen had given
prominence were definitely geometrical in character and these
continued to be studied. A new interest was injected into these
somewhat academic studies by Bartholinus's discovery of double
refraction in i66g. This was described in a short Latin treatise,
Experiinenta Cryslalli Islandici Disdiaclastici . . . , which was
widely read. Huygens obtained a specimen of Iceland Spar and
a considerable part of his Traite de la Lumiere, completed in
1678 (published 1690), deals with experiments he then carried out.
Huygens had developed a pulse theory of light and the effort
he made to reconcile his theory with the new and peculiar optical
properties of Iceland Spar was a masterly one. Curiously enough
there is little reference to this work in his correspondence.

About this time Huygens had as his assistant Denis Papin, a
Frenchman who later worked with Boyle in England. With
Papin, Huygens in 1673 experimented on gunpowder as a source
of mechanical energy. There is a possibility that Huygens had
considered some kind of atmospheric engine as early as 1660
when he talked with Pascal about " the force of water rarefied
in cannons ". In thfcse experiments of 1673 we can see the fore-
runner of Papin's atmospheric engine, which did in fact employ
steam in place of gunpowder. Papin was later, through the in-
terest of the Landgrave of Hesse, appointed professor at the
university of Marbourg and it was here that he developed the
atmospheric engine which gave Newcomen his clue.

In exchange for Papin, as one might say, Oldenburg sent over

THE LIFE OF CHRISTIAN HUYGENS 75

to Huygcns the wealthy young amateur Walter von Tschirn-
haus, a friend of Spinoza and Leibnitz. He belonged to a class
which had early supported the new scientific societies but he was
exceptional in that his interest in science did not quickly flag and
in that he made himself something more than a mere dilettante.
Through Tschirnhaus Huygens undoubtedly learned more of
Spinoza's philosophical ideas but he showed himself little inter-
ested in them. Unlike many of the seventeenth century men of
science Huygens did not occupy himself with philosophical or
theological questions and neither he nor Leibnitz seems fully to
have grasped the nature of Spinoza's thought.

XII

Early in 1676 Huygens was again ill. There is no doubt that
the illness of 1670 had recurred and this time he showed greater
caution in meeting the danger. In March 1676 he set out to
return home to the Hague while he was yet able, but the journey
was slow and very uncomfortable. To his brother he confessed his
doubts whether he would return to a life in Paris which seemed
to be injurious, and even when he had recovered, a year later,
he procrastinated under the pretext of uncertain health. Colbert
gave permission for his remaining at the Hague for the winter of
1677-78 and the return journey to Paris took place in June 1678.

During these two years at home he pushed on with his re-
searches. To these years belongs a great deal of work on the
double refraction of Iceland Spar and the development of his
wave theory of light. On November 122, 1676, Roemer read a
paper to the Academic Roy ale in which he gave the first calcula-
tion of the velocity of light. Huygens was immediately
interested on receiving a copy of the paper and an interesting cor-
respondence with Roemer was begun. The assumption that light
travelled with a finite speed was fundamental in Huygens's work
and on this assumption, he wrote to Colbert, he had " demon-
strated the properties of refraction and, a little while since, that
of Iceland crystal which is no small marvel of nature nor one
which it is easy to understand ". It was consequently gratifying
that this assumption should receive confirmation and that the
speed of light should be approximately known. There was some
similarity of outlook between Roemer and Huygens for Roemer
supposed, like Huygens, that the passage of light through cry*

76 THE LIFE OF CHRISTIAN HUYGENS

tals (thrown into prominence by the Iceland Spar phenomena)
was analogous to the transmission of impulses through con-
tiguous spheres. The explanation of double refraction along these
lines would, he considered, establish the truth of the theory of
light which for some years had been associated with Huy gens'?
name.

It is well known that Huygens was led to his theory of trans-
mission of light through his work on impact. The transmission
of longitudinal, compressional, vibrations through perfectly
elastic spheres seemed to him to have applications in light, since
crystals and other transparent media might be supposed to be
composed of assemblages of atoms. And even though he was
unable to suppose the atoms of the elements were the actual
medium since all substances are not transparent he found a
mechanism which reduced light to a form of motion and brought
it within the treatment of the " true Philosophy ". In this philo-
sophy " one conceives the causes of all material effects in terms
of mechanical motions. This, in my opinion, we must necessar-
ily do, or else renounce all hopes of ever comprehending any-
thing in Physics/ 1 This quotation is taken from the beginning
of Huygens's Traite de la Lumiere. To explain the transmission
of light through the Torricellian space and all manner of trans-
parent substances, however, some pervading subtle medium was
assumed. We must remember that Huygens was led to conclude
that such a medium existed from his vacuum experiments. The
ether was, however, not a continuous medium but was composed
of very light particles in contact. These, on impact with the
heavy vibrating atoms of incandescent bodies, transmitted their
vibrations in all directions according to the laws of impact. The
elasticity of air, Huygens thought, " seems to show " that it
is made up of particles which are " agitated very rapidly in the
ethereal matter composed of much smaller parts ". It was im-
portant that slight impulses travelled as fast as strong ones, a
fact which was readily explained by applying Hooke's law
of elasticity to the particles of ether. Also, individual wave-
lets by themselves were too weak to produce effects of light,
which only arose when the wavelets combined to form a wave-
front according to the well-known Huygens construction now
given in all text-books on light.

Huygens's theory is better described as a pulse theory rather
than a wave theory but in the Traite he made the remark that

THE LIFE OF CHRISTIAN HUYGENS 77

the vibratory motion "... is successive and . . . spreads as sound
does, by spherical surfaces and waves ", Hooke developed a wave
theory also largely in relation to his observations of colours
produced by thin films. The difference between his ideas
and those of Huygens was mainly that Hooke did not consider
the formation of a wave-front by the innumerable individual
wavelets.

Not all scientists, however, were prepared to accept Roemer's
estimate of the velocity of light. Descartes had been so con-
vinced of the instantaneous transmission of light that he un-
wisely said he would stake all his system of philosophy on its
truth. Unlike Huygens there were many who remained under
his spell. In the Traite, therefore, Huygens went to some pains
to demonstrate the error of Descartes's reasoning. Cassini was
opposed to Roemer's explanation of the apparent advance and
retardation of the occultations of Jupiter's satellite, but mainly
because only the innermost satellite had been studied. When the
Academic had to decide on the dispute which arose over the
work they came to the conclusion that Roemer was right; as he
explained, the occultations of the outer satellites were less fre-
quent and less sharply observable for obvious reasons. While the
method he had put forward was the best one available for finding
the velocity of light, he hoped that surface marks on Jupiter
would prove of use, and later in the year observation of a spot
on the planet gave the period of revolution of the planet on its
axis. Observation of this spot could then be relied on in place of
occultations for measurement of the velocity of light. Clearly
Roemer was a man of the same outlook and ingenuity as Huy-
gens. At the time when Huygens was at the Hague, Roemer was
attempting to determine what effect the motion of the earth
should have on the apparent positions of the heavenly bodies
when this motion was transverse to the direction of the light
rays. It scarcely matters that Roemer conceived the problem in
terms of the Cartesian vortices; the point was that the circular
motion of the terrestrial vortex should produce an apparent cur-
vature of the path of light. In its modern form the problem
was propounded and explained by Bradley, who discovered the
effect of " aberration" in 1728.

Huygens made the journey back to Paris in the middle of the
summer of 1678. With him went Nicholas Hartsoeker, later
known as a maker of lenses. Once more Huygens settled down to

78 THE LIFE OF CHRISTIAN HUYGENS

his old occupations. The period to which we have come was,
unhappily, one during which he was not for long well. He was
ill again in 1679 an d although he recovered he was compelled
again to return to the Hague in 168 1 . From this last convalescence
he never returned to Paris. We are consequently faced with the
fact that this was Huygens's last stay in Paris, and one which was
seriously interrupted by illness.

Curiously enough, in view of the swift reverse which was in
store, Huygens's prestige seems never to have been higher than
it was at this time. It is clear that he was widely regarded as in a
real sense the head of the Academic Royale des Sciences, the
position of which seemed even more assured than that of the
Royal Society at this time. The Royal Society in fact suffered
from the political upheavals of the time and from the defection
of some of its members. 1678 was the year of the Popish Plot,
which, according to Titus Dates, aimed at the conquest of the
kingdom by the Jesuits. As late as November it was held that
" there hath been and still is a damnable and hellish plot, con-
trived and carried on by popish recusants, for the assassinating
and murdering of the King and rooting out and destroying the
Protestant religion ". With the death of Oldenburg in this year,
Huygens's relations with the society were practically at an end.
The Academic Royale had on the other hand increased in
vigour after its slow development in the first few years. Thanks
to Colbert, Huygens and Auzout had been able to equip the
Academic with all the laboratory and astronomical apparatus
required and each year saw improvements in scientific technique.
The society had begun to undertake enterprises such as the ex-
pedition of Richer to Cayenne in 1672 and this had led to impor-
tant information concerning the shape of the earth.

Nevertheless, in France as in England there were jealous
opponents of the new learning. The universities always feared
an undermining of their authority if the scientific societies be-
came too strong or too serious in their tasks, the Jesuits wished
to have a monopoly of the new knowledge, and there were some
who stirred the popular mind against investigations which
seemed to a less and less degree to aim at the production of new
inventions or the amelioration of life. In France, Paul Pelisson,
who was writing a history of Louis's reign, gave Huygens space
to deal with current criticisms and to enlighten the public on the
aims and work of the Academic.

THE LIFE OF CHRISTIAN HUYGENS 79

In this review Huygens limited himself to the particular field
in which he was an authority. He began by alluding to the need
of astronomical studies and the great importance of the new
observatory. The use of pendulum clocks and improved tele-
scopes had made possible observations incomparably more exact
and more easy than they were in the time of Tycho Brahe. The
discovery of new stars, the confirmation of the ring of Saturn and
the discovery of its satellites, a full study of the moon's surface
and the description of comets and sunspots were among modern
achievements. The discovery and measurement of the velocity of
light were adduced as a consequence of such observations. Nor was
such work without practical value : the occultations of Jupiter's
satellites provided a method of determining longitudes, a problem
for which the pendulum clocks might soon provide a better
answer. The great appeal of the work of the Academic, however,
lay in the steady expansion of man's knowledge and understand-
ing of the world. Expeditions had been sent to Cayenne and to
Hveen; more exact star catalogues and ephemerides were to be
prepared so that the theories of the universe might accord more
exactly with observation. The Earth itself had been made the
object of scientific measurement. Geometry had been applied " in
the study of causes in the field of Physics, it being accepted by
almost all philosophers of today that the figure and movement
of the corpuscles of which everything is composed are alone re-
sponsible for all the wonderful effects which we see in nature ".
This is really a statement of the new mechanistic philosophy to
which the physical researches of the men of science had led.
The world had come more and more to be regarded as a perfect
machine and, says Burtt 1 , "first in Huygens and (in a more philo-
sophical form) in Leibnitz we have this opinion unequivocally
proclaimed ". Huygens clearly feared that this summary might
seem to be written from the standpoint of the vortex theory of
Descartes and he went on: " which opinion having been greatly
supported through the philosophy of Descartes, they neverthe-
less adhere neither to his sentiments nor to those of any other
philosopher in order to gain authority ". Descartes, he pointed
out, was mistaken in many things through lack of experiment
and in particular he had sacrificed the accurate definitions of con-
cepts which Galileo had begun to clarify. Truer ideas were now
held regarding motion and force and momentum, the nature of

1 E. Burtt: The Metaphysical Foundations of Modern Science (193*).

8o THE LIFE OF CHRISTIAN HUYCENS

meteors and other celestial phenomena, and the nature and
effects of light. Microscopes, telescopes, the air pump and many
other machines had been brought into use and had extended
man's knowledge and led to the advancement of the sciences.

Nevertheless, while Huygens was an enthusiast for the new
world of the seventeenth century science, he displayed caution
in comparing his own age with that of classical antiquity. His
friend Charles Perrault frankly regarded the seventeenth cen-
tury as superior to all other ages and considered Huygens him-
self an example of this superiority. To such praise and to that
of the younger Fermat, who compared him with Descartes, he
returned a modest reply. " I am one of those who have profited
from the wisdom of that great man," he wrote.

That Huygens was not everywhere so popular and that there
were factions in the Academic at this time can scarcely be
doubted. The eminence of Huygens, in whose honour a medal
was struck in 1679, was not agreeable to Cassini and de la Hire
and the latter is known to have led an opposition to the entry of
all foreigners and especially the friends of Huygens into the
academy. In France just as in England religious differences were
being exploited for political ends. The greatest division within
the Academic Royale seems to have resulted not from national-
istic or religious partisanship but arose between members who,
like the original Montmorians, were eminently followers of Des-
cartes, and those who, like Huygens and Mariotte, showed an
increasing scepticism towards the Cartesian system. In these cir-
cumstances Huygens seems to have felt more affinity with men
of science who were not involved in the dispute and with Leibnitz
in particular there grew up an interesting correspondence.

It will be remembered that Leibnitz had studied mathematics
with Huygens in 1672. During the subsequent years the German
mathematician had pursued his researches along new paths. In
1676 he had been in correspondence with Newton about methods
of expansion in series. Newton mentioned his binomial theorem
and the method of fluxions but did not describe the latter,
although he added some illustrations of its use. By 1675 Leibnitz
was employing his own form of differential calculus but was un-
able to involve Newton in discussing anything which might
arouse controversy. Newton's method of fluxions was, in fact,
not published until 1693. In correspondence with Huygens, Leib-
nitz claimed to have developed the calculus into a method by

THE LIFE OF CHRISTIAN HUYGENS 8l

means of which he had successfully treated a variety of prob-
lems. Huygens, however, would not abandon geometry for the
differential calculus and never gained any facility in its use. Leib-
nitz wrote at length also about the subject of symbolic logic, of
which he was an originator, but his ideas were not appreciated by
Huygens or anyone else at that time and they wer6 not taken
up until the following century. Leibnitz was anxious to secure
nomination to the Academic Royale as a foreign member but
this Huygens seems to have been unable to obtain. Not until
1700 was Leibnitz and in the same year Newton admitted as
a foreign member. It was in 1700 that Leibnitz organized the
Berlin Academy of Sciences.

At the Academic Royale Huygens's chief activity at this
time was the presentation of his work in geometrical optics in
a series of lectures lasting from May to August. The whole sum-
mer of 1679 was spent in editing the work of many years before
and with special problems concerning Iceland spar. Much of
the work on the optical properties of conic sections which comes
at the end of the Traite de la Lumiere was completed about this
time. Fermat's least-time principle, also, he succeeded in de-
ducing for refraction on the assumption that light travels more
slowly in glass or water than in air. As for Descartes, both
Huygens and Leibnitz had scant regard for the greater part of
his work in this field. His " pretence of a demonstration " of
the laws of refraction was replaced by the well-known treatment
which employs Huygens's secondary wavelets. Huygens's work
in optics may in fact be regarded as standing in relation to
previous studies by Kepler, Snell, Descartes, and Fermat much
as Newton's mechanics stands in relation to the mechanics of
Galileo and Huygens : Huygens achieved the same union of the
physical and mathematical aspects of the subject. His mastery
of geometry of course equipped him in a unique way for this
task. The subject of colour was, however, left on one side;
Huygens seems always to have held that a mathematical ex-
planation of this was not possible. Nevertheless, he appreciated
the practical outcome of Newton's work in this subject: the
discovery of chromatic aberration of lenses showed, he saw, that
this effect may be no less important in telescopes than spherical
aberration. It followed that the search for the achromatic lens
might be more profitable than the attempt to obtain lenses with
non-spherical surfaces had been. He would probably have had to

82 THE LIFE OF CHRISTIAN HUYCENS

admit by this time that the idea of the reflecting telescope was
by no means " extravagant ".

The interest aroused in microscopic work by Hooke's Micro-
graphia and the translation from Dutch into French of
Leeuwenhoek's work by Huygens about 1677 led many at this
time to take up such studies. The whole world of infusoria
awaited discovery. The imperfections of the compound micro-
scope were still considerable, however, and Leeuwenhoek, as is
well known, preferred to use a single lens of short focal length in
his observations, which perhaps included the discovery of bac-
teria. Huygens used very small lenses of glass, some of which
he made hollow and filled with alcohol. Locke, who was in Paris
in 1678, wrote to Boyle of the " extraordinary goodness " of
Huygens's microscopes. In devising a mount for his very small
lenses Huygens introduced a method of altering the intensity of
illumination of the object. Later, in 1692, he introduced dark
ground illumination. These were the contributions of a prac-
tical microscopist. After 1676, in fact, Huygens was very
interested in making observations of infusoria in rain water.

Ill-health no doubt accounts for a diminution of the mathe-
matical and more abstract studies of Huygens after 1680. He left
Paris at the end of the summer of this year for a short stay at
Viry, where the country air restored him for a time. He returned
to the capital in time to take part in observations of a comet and
as a recreation started the construction of a planetary machine
which would reproduce by means of clockwork the relative
motions in the solar system. Early in 1681 he was again ill, but
not until September was his return to Holland practicable.

XIII

The convalescence after this last illness was slow. Letters
arrived bearing the good wishes of men of science in Paris and
London. Even de la Hire, only recently elected but before long a
prominent member of the Academic, sent the good wishes of
" all the company ". It is evident from his letter that de la Hire
hoped for the position which Huygens had left at least tempor-
arily vacant; there is a strong presumption that he did in fact
work hard to prevent Huygens from having much opportunity
to return to Paris. Huygens for his part was at first in no hurry
to leave Holland even when, in 1682, he had practically

THE LIFE OF CHRISTIAN HUYCENS 83

recovered. Later in the year the Dutch East Indies Company
showed interest in the latest pattern of marine clock and this
was a further incentive to remain. By taking up the summer of
1683, the work on the new marine clock really decided his future,
for Colbert died in September of this year and without his
patron's support the opposition to Huygens's return began to
be really formidable. Not only this but the political horizon was
dark with the uncertainties caused by the renewal of Louis's
activities abroad. By 1686 the situation in Europe was not unlike
that of 1673. The Revocation of the Edict of Nantes (1685)
roused all Protestant countries. To Holland fled a large number
of exiled Frenchmen. It was a time of rapidly diminishing free-
dom in France and Huygens's experience well illustrates the fact.
For when he renewed his application to return the appeal fell on
deaf ears. Whether anti-Protestant feeling was the sole reason
is not altogether clear. Huygens's friend Roemer left Paris a few
months earlier, and it was four years later that the Edict was
repealed. Quite as much as anti-Protestantism, probably, per-
sonal jealousies spoiled the work of the Academie; the years
after 1681 seem indeed to have been years of retrogression

It is a striking illustration of the hostility which the
Cartesians had come to feel for Huygens at this time that the
Abb Catelan should, nine years after the publication of the
Horologium Oscillatorium, attack the fundamental principles
employed by Huygens in his treatment of the compound
pendulum. There is nothing of scientific interest in Catelan's
criticisms and they were designed to discredit Huygens's work
in the eyes of those who were comparatively ignorant of mathe
matics and mechanics. The mathematician James Bernoulli
came forward to champion Huygens's ideas in 1684.

In the middle of 1684 Huygens was visited by Thomas
Molyneux, a contemporary and acquaintance of Flamsteed and
Hooke. Molyneux wrote to his brother that he was received
" extraordinarily civilly ". Huygens, he said, " beyond my ex-
pectations talked to me in my own language, and pretty well ".
He was shown Huygens's planetary machine which he decided
was " nothing more than an ingenious curiosity " for, he said, " I
asked him could he by help of it exactly determine an eclipse,
and I observed that he could not give me a positive answer, as
being loath to confess the imperfections of his contrivance to me
that seemed to admire it so much as I did ". Huygens had in fact

84 THE LIFE OF CHRISTIAN HUYGENS

come to the end of his great period of scientific activity, but
some profound ideas were yet to be developed. The news of the
death of Picard (1682), who had done notable work at the Paris
observatory, caused Huygens to ponder the uncertainty of life
and to think of publishing the " many good and useful things
which I have written or found, to complete which I desire only
peace and the continuance of my health ". As events turned out
his retirement held more of solitariness than he desired. The
death of his father at a great age in 1687, and the departure of his
brother Constantin for England in 1688, when William of
Orange became King of England, left him alone in the isolated
residence at Voorburg in the summer. The winters he spent at
the Hague. In his letters he lamented the absence of any with
whom he could discuss scientific topics. Owing to financial
worries he began to consider the possibility of securing a position
as counsellor to William III, but this only embarrassed the King,
who perceived that Huygens had " higher ideas than to loiter
with administrators ".

The idea of a position in England seems to have occurred to
Huygens after a short visit to this country in 1689. He was in
London from June to August of this year, but only brief records
remain. He met Flamsteed at Greenwich and attended a meet-
ing of the Royal Society at Gresham College. In company with
Fatio de Duillier, a Swiss mathematician, he met Newton for the
first time. Little is known about this or of another occasion
when, in July, Huygens, de Duillier, and Newton travelled from
Cambridge together on the occasion of Newton's application
for the position of Provost of King's College. Huygens also met
Boyle on several occasions and witnessed some chemical experi-
ments. He left London with many regrets for the isolation in
which he then lived.

It would, of course, be extremely interesting to know what
discussions took place between Newton and Huygens on the
occasions when they met. It is clear that in mechanics the two
scientists held certain divergent views, notably on the subject of
the conservation of energy and on the existence of absolute space
and time. For Huygens, after reading the Principle, became
strongly critical not only of Newton's postulate of universal
gravity but also of his belief in the existence of absolute space
and motion. He had early perceived that a body, moving uni-
formly in a straight line with respect to one observer, might be

THE LIFE OF CHRISTIAN HUYGENS 85

accelerated with respect to another. And while he first made a
distinction in favour of the absolute nature of motion in a circle
which was accompanied by the existence of a centrifugal force,
he abandoned this also after reading the Principia. This was in
contrast with Newton's acceptance of an absolute space and time
according to which all motion possessed an absolute character.
Whether they discussed such differences of view and whether
they compared notes on the subject of resisted motion and other
matters in which they were both interested at this time is not
known.

Over the question of the cause of gravitational attraction
there was, of course, a complete divergence of view between
Newton and Huygens, for while the latter speculated for some
time on the subject it was one for which Newton felt no interest.
Huygens went out of his way to expound his views in such a way
that they would not give Newton any offence. He seems to have
been a little nervous about Newton's reception of yet another
hypothesis. It may be remarked here that Huygens's theory
grew out of his work on the nature of light and was an attempt
to explain gravity as due to the action of an ether or fluid matter
which, owing to rotation, seeks to travel away from the centre
and thus, as he thought, forces slower moving bodies together.
Even at this time, however, the difficulties of such a theory were
becoming clearer. A fluid which could permeate matter could
scarcely exert a reaction on it and de Duillier, who had gone to
England for the purpose of studying Newton's works, pointed
out to Huygens that the absence of any apparent resistance to
the motion of planets and comets argued that the ether must be
excessively attenuated. As is now known, however, Newton was
not so thoroughly opposed to the ether theory as was generally
supposed. Although he condemned the idea (as expressed by
Hooke) in 1675, he returned to the question in the " Queries " to
his Optics.

The inverse-square law of gravitational force posed great
difficulties for Huygens's mechanistic theories. It was, he said,
" a new and very remarkable property of gravity of which it
was very necessary to search out the reason ". He could not see
that the cause could be given on the principles of mechanics or
of the rules of motion. The view that gravity was an inherent
property of matter, he said, "takes us very far from the
principles of mathematics or mechanics ". Leibnitz also was

86 THE LIFE OF CHRISTIAN HUYGENS

against Newton's view of gravity as innate. If it was to be
regarded as a " law of God who brings about this effect without
using any intelligible means, then it is -a senseless occult
property, which is so very occult that it can never be cleared up,
even though a Spirit, not to say God himself, were endeavouring
to explain it," he wrote to Hartsoeker.

The importance of Newton's work was not, however, lost on
Huygens, who perceived that the Principia utterly destroyed the
Cartesian vortices. Writing to Leibnitz about the elliptical orbits
of the planets Huygens said he would like to know if he could
continue to hold to Descartes's vortices after reading the
Principia since these " in my view are superfluous if one accepts
the system of Mr. Newton in which the movement of the planets
is explained by the gravity towards the Sun and the vis centri-
fuga which balances it ... ".

The extreme Cartesian view of gravity was expounded (1690)
by Regis in a book on Richer's observations at Cayenne. The
explanations put forward were closely similar to the ideas ex-
pressed by Huygens in 1669 on the occasion of the discussion at
the Acad&nie Royale. Regis made no mention of Newton in his
book. In 1690 Huygens felt a good deal of uncertainty and
wavered between his original ideas and the view expressed in his
letter to Leibnitz. The appearance of the tract on the cause of
gravity at the end of the Traitf de la Lumtire, published in this
year, cannot be held to represent Huygens's final views, about
which more will be said later. In England the effect of the
Principia was more profound. Fatio de Duillier said that some
of the Royal Society were "extremely prepossessed" in the
book's favour and reproached those who were not under its spell
as being too Cartesian. " They . . . have led me to understand
that after the meditations of their author all Physics has been
much changed " lie wrote to Huygens. There can be no doubt
that on the Continent the criticisms made by Huygens and
Leibnitz strengthened the position of the Cartesian philosophy
for a good many more years. And yet Huygens's own work was,
at its best, as opposed as Newton's to the Cartesian frame of
mind and he did a great deal to dispose of the errors of
Descartes's physical ideas, llie last five years of Huygens's life
were in fact to be years of crisis for the Cartesian philosophy.
Leibnitz and Huygens would have developed an alternative
analysis which freed itself from Descartes's errors while at the

THE LIFE OF CHRISTIAN HUYGENS 87

same time rejecting Newton's conception of matter, time and
space but this project was never carried through. In the event
their effect was to delay the acceptance of Newton's work while
at the same time weakening the supports on which Descartes's
physical teaching rested.

In the meantime the more orthodox Cartesians were driven to
great lengths to show that the new results of scientific research
were fundamentally in accord with Descartes's ideas. Even the
differential calculus was claimed by Catelan to be derivable from
Descartes's geometry. The faulty treatise which he produced to
support his view led to a dispute with the mathematician de
I'Hdpital. The latter, regarding Huygens as a natural ally, gave
violent support to the latter's mechanics, also criticized by
Catelan. This somewhat embarrassed Huygens, who was by no
means sure of some of de I'Hopital's ideas on this subject. For
de l'H6pital tried to obtain some sort of proof of the principle
that the centre of gravity of a system of connected bodies cannot
rise under the sole action of gravity. Huygens preferred to
regard this principle as self-evident. Pascal and Torricelli, he
pointed out, had used the same idea though limiting it to
statics.

Of a different character was Huygens's correspondence with
Pierre Bayle about this time. This famous French sceptic was
appointed professor of philosophy at Rotterdam in 1681, so that
he arrived in Holland in the year that Huygens returned from
Paris. Under the conditions of Catholic intolerance the intellec-
tual ferment, once concentrated in Paris, was becoming diffused
into the freer but less educated provinces and into Holland. In
1684 Bayle started a periodical entitled Nouvelles de la
r^publique de lettres, the first number of which he sent to
Huygens. The latter became interested in Bayle's aims and
received him at his house, where he enlightened him on the
subject of scientific studies. His correspondence with Bayle came
to an end, however, after the philosopher was condemned as an
atheist in 1693. Bayle's view was that religious dogma is of its
nature irrational and that there is no merit in Relieving that
which is merely consonant with reason. This outlook of credo
quia absurdum was one which could not appeal strongly to
Huygens.

Rather more interesting was Huygens's correspondence with
Pierre Daniel Huet, another sceptic whose avowed purpose was

88 THE LIFE OF CHRISTIAN HUYGENS

to discredit reason in the realm of religious belief. Both Huet
and Bayle actually influenced thought in the opposite direction
to that which was intended, that is, towards scepticism. Huet in
addition was strongly opposed to the rationalism of Descartes
and strongly in favour of empiricism. In October 1689 he sent
Huygens a copy of his Censures on the Cartesian philosophy. In
reply Huygens said that he also had meted out rough treatment
to Descartes, and that he hoped that his scientific work had
replaced Descartes's doctrines with truer ideas. He agreed with
Huet that while Descartes had overthrown the older philosophies
he had borrowed from them their dogmatic spirit. He had had
the ambition to be the author of a new philosophy and in his
haste had been led to maintain ideas even against disproof. To
Huygens this philosophy appeared as the successor of Aristotel-
ianism. Nevertheless, when Martin van Helden, a Cartesian and
professor of mathematics at Louvain, was threatened with im-
prisonment for his criticisms of scholastic philosophy, Huygens
assisted him so that he should not become "a martyr to
Cartesianism ". He did not feel very strongly over this matter, for
the battle of experimental science against the a priori philoso-
phies seemed to him to be won. " It seems to me " Leibnitz was
able to say, " that the Cartesians have very much declined and
that they have not too many able men ".

XIV

In 1685 Huygens was still negotiating over his return to Paris
and there were endless letters, many of them unanswered* sent
off from Holland. Nevertheless, it is not really certain that he
wanted to go back, and he may well have been deterred by know-
ledge of the changed conditions at the Academic. Up to 1688
he stayed on at the Hague, and in the spring of that year he
settled at Hofwijk, a property in the neighbourhood of the city
which had belonged to his father. After his father's death in
March 1687, the house was lent to Christian by his brother
Constantin, who left with William III on his memorable expedi-
tion to England in the following year.

In these last years (1685-95) Leibnitz was solicitous about
Huygens's unpublished works and recommended him to con-
serve his strength, for, he wrote, " I do not know anyone who
could replace you ". Huygens's old age was a lonely one and he

THE LIFE OF CHRISTIAN HUYGENS 89

was troubled with ill-health, but, he replied, " I see that one
becomes accustomed to all these things ". He was not quite for-
gotten, for Charles Perrault drew public attention to the great
scientist's work in his Parnllele des Anciens et des Modernes and
when Varignon was about to publish a book on mathematics he
felt that he must take the opportunity, as he said, of paying
homage to " the greatest mathematician of our age ".

The pattern of Huygens 's life remained much the same right
up to the end. He continued to work on the improvement of
lenses, on the spring-regulated clock and the marine clock, and
the writing of his last work, the Cosmotheoros. Undaunted by
the unpromising performance of the various marine clocks he
had constructed since 1663 he continued, with characteristic
patience, to labour at this still urgent problem of the marine
chronometer. In 1685 he went himself on a short trial (the only
one) on the Zuyder Zee. In 1686 and 1690, clocks fitted with
bifilar pendulums were sent in charge of the captains. All these
trials were unsuccessful. The failure of the bifilar pendulum was
the greatest disappointment, for work on this type of clock dates
as far back as 1673 at least, since it was described in the
Horologium Oscillatorium. After 1690 Huygens experimented
with a new type of regulator and reverted to the spring drive
which he had tried at the beginning and then abandoned. The
new clock went well in laboratory trials and in 1694 Huygens
hoped that the Dutch East Indies Company would take it up. He
died before anything further could be done.

The fact that Huygens could not accept the chief conclusions
of Newton's Principia is the most interesting fact that comes out
of his correspondence at this time. Five years after its appear-
ance he wrote of Newton, " I esteem his understanding and
subtlety highly, but I consider that they have been put to ill use
in the greater part of this work, where the author studies things
of little use or when he builds on the improbable principle of
attraction." The idea of universal gravitation " appears to me
absurd " he wrote. Yet he felt compelled to admit that Newton's
explanation of comets was incomparably better than anything
imagined by Descartes. It was difficult to see how comets could
cut across the vortices imagined by Descartes, or to explain the
eccentricity of the planetary orbits and the real accelerations and
retardations of the planets in their orbits except on the lines laid
down by Newton. Over the shape of the Earth, also, Huygens

90 THE LIFE OP CHRISTIAN HUYGENS

was in accord with Newton. He did not deny that if the gravita-
tion of the planets towards the sun were taken as inversely
proportional to the square of their distances " this, with the cen-
trifiigal virtue gives the Eccentric Elliptics of Kepler ". But he
and Leibnitz, far from feeling that this reduced the solar system
to order, felt that it raised an insistent question of how gravita-
tion arose. Leibnitz thought he could perceive an analogy with
the intensity of light which, as a simple geometrical deduction,
also obeyed the inverse-square law. Rays of attraction might be
imagined which caused bodies to descend if their centrifugal
force diminished. These rays were dismissed by Huygens, how-
ever, as incompatible with his theory of a circulating medium.
It almost seemed as if a return might be made to Kepler's identi-
fication of gravity with a kind of magnetic attraction. Leibnitz,
at least, inclined not a little to this view. Both he and Huygens
insisted on attributing the effects of gravity to the medium
which they believed pervaded the universe. Consequently they
were both interested in the study of motion in a resisting
medium, for they no doubt perceived that this was the Achilles'
heel of their system. If the medium had mechanical properties
exhibiting themselves as gravitational force, magnetic force and
in other ways, what influence must it have on the orbital motions
of the planets and on terrestrial motions? Newton's Principia
had dealt with this problem and much of the work was deliber-
ately aimed at the overthrow of the Cartesian vortices. Huygens
considered Newton's treatment to be not without fault but he
agreed with him as against Leibnitz over the definition of resist-
ance, " for you," he wrote " call the resistance the velocity lost
or the loss of velocity caused by the medium . . . For Mr.
Newton and myself, however, the resistance is the pressure of the
medium against the surface of the moving body . . . " It is
really astonishing to us now that Huygens did not see that
Newton's study of resisted motion completely disproved the
vortex theory, but we must remember that the elastic fluid theory
was held in the nineteenth century under even greater difficul-
ties. Furthermore, a comparison of Huygens and Leibnitz at
this date leads to a decision in favour of Huygens's notions. In
1692 Leibnitz still supported vortices, while accepting Kepler's
laws; Huygens had at least got to the point of seeing the over-
whelming force of the quantitative work of Newton even while
he rebelled against innate gravity. As he finally left them, the

THE LIFE OF CHRISTIAN HUYGENS 91

vortices were considerably attenuated affairs, suitable only for
popular exposition of the sort found in the Cosmotheoros.
Leibnitz, on the other hand, converted the subtle matter of
Descartes into a production of his own : the materia ambiens.

A good deal of what we should term pure mathematics
crops up in Huygens's and Leibnitz's letters. Leibnitz took up
several of the problems studied by Huygens and gave them
new form. " My design has been " he wrote, " to give a little
trouble to these good Cartesians who, through having read the
Elements of Bartholin or Malebranche, believe they can do all
in Analysis." There followed a series of letters in which Leibnitz
gave Huygens an account of the differential calculus and its
uses. He was able to investigate the properties of a curve like the
cycloid, he said, from a purely analytical treatment and without
any recourse to the figure. In regard to the calculus, Leibnitz
was not a clear expositor. It is clear that from one aspect the new
method was regarded not so much as a development of pure
mathematics as an instrument for physical research. The union
of mathematics with experiment is what Sir William Dampier
has called the " new mathematical method ". For Huygens, as
for Galileo and indeed for Newton, experiment had not achieved
the position it later held in certain branches of science. From a
comparatively few observations, by the aid of " geometry " one
could advance far into new realms, a fact which is well illustrated
by Huygens's work on impact and on the compound pendulum.
" It must be admitted " wrote Huygens, " that geometry is not
made for all sorts of minds."

From being sceptical Huygens soon became envious of the
calculus differentialis. Finding Liebnitz's accounts rather
obscure he wished that either he or Bernoulli could be there to
assist him. Some collaboration did indeed spring up with Fatio
de Duillier, and Huygens's note-books contain many pages of
working on the new lines. The great change of outlook was a
difficult one for the great geometer and he did not attain facility
in the use of the calculus. The new calculus, Leibnitz
emphasized, gave its results by a kind of analysis without any
effort of the imagination, " and it gives us over Archimedes all
the advantages which Vieta and Descartes have given us over
Apollonius ".

Fatio's work on the calculus is important in the history of the
subject, for it was through him that the dispute between the

92 THE LIFE OF CHRISTIAN HUYGENS

followers of Newton and Leibnitz sprang up. More's Isaac
Newton (1934) gives a good account of the episode. Fatio seems
to have become resentful of Leibnitz's rather superior criticism
of his work, which was of an undistinguished nature, and it is
considered that Fatio smarted under a sense of grievance. After
returning from England, where he had been in contact with
Newton, Fatio wrote to Huygens, saying that priority for the
invention of the differential calculus certainly belonged to
Newton. He suggested that Leibnitz's ideas were in fact obtained
from Newton's letters which went back to 1676 and 1677. The
publication of these, he hinted, would embarrass Leibnitz.
Actually, these dates were beside the point since both mathe-
maticians seem to have used the calculus much earlier. Leibnitz,
at any rate, affected to be unimpressed by news of Newton's
advances and hinted to Huygens that he had done a number of
things of which Newton knew nothing. Huygens's part in the
quarrel which ensued was small but noteworthy for it was
through him that Leibnitz first learned of the charges made by
de Duillier.

About this time an interesting comparison in mathematical
methods was made through the study of the same problem by
Leibnitz, Huygens and James Bernoulli. This problem was the
one propounded by Mersenne many years before; to find the
theoretical form of a chain suspended from its two ends which
are at the same height, so that a curved line hangs between them.
The publication of the results showed a fair agreement between
the three mathematicians but showed up the advantages of the
calculus, which was more and more applied, not only to new
problems but also to others already solved by classical methods.
Huygens was not altogether pieased by some of the new
methods, notably that adopted by James Bernoulli in solving the
problem of the centre of oscillation. But the truth is that the
form of Huygens's work in mathematics had always been some-
Wjhat reactionary and was fast becoming obsolete.

There is a faint echo of the Royal Society and the pleasant
days spent in England in some of Huygens's last correspondence.
Constantin, his brother, in 1691 presented a telescope objective
having a focal length of 1-22 feet to the society. This was by no
means the best achieved by Huygens. During these years he
made one of 210 feet focal length. The 122 foot lens was a fine
objective, however, and Hooke was entrusted with the erection

THE LIFE OF CHRISTIAN HUYGENS 93

of an aerial telescope to accommodate it. The lens was marked
by Constantin so that it should afterwards be readily identified
Constantin apparently not finding Hooke any more trustworthy
on personal acquaintance than he had anticipated. This mark-
ing has made possible the identification of the objective and
examination of its quality in recent times. The figuring and
centring of the surfaces is described as "astonishingly efficient,"
but the quality of the glass is " hopelessly bad ". From
Constantin, Huygens heard of the death of Robert Boyle. " Mr.
Boyle is dead as you know already without doubt " he wrote to
Leibnitz, and added " It seems strange that he built nothing on
all the experiments with which his books are filled; but the thing
is difficult, and I never thought him capable of as great an
application as is necessary to establish the true principles."
Since Huygens had often expressed his admiration of Boyle this
comment may be taken to indicate more esteem of the difficul-
ties of Chemistry than depreciation of the father of that science.

The other prevailing practical interest, besides the clock, to
which Huygens gave attention during these last years was the
telescope. In 1684 he published his Astroscopia Compendiaria.
This contained an account of tubeless telescopes and this may
explain how it is that Huygens has been supposed to be the
originator of this difficult and ultimately unsatisfactory tech-
nique. He was compelled to reconsider his attitude to the reflect-
ing telescope through the apparently insuperable obstacles raised
by aberration chromatic and spherical. Yet he still preferred
the refracting telescope because metal mirrors took such a poor
polish and the grinding of glass mirrors was extremely difficult.
Moreover, it was at that time very difficult to silver the upper
surface and silvering the back surface meant that a second image
was formed by partial reflection at the top surface so that a
double image was formed. Even if the radii of curvature of the
two surfaces were such that the two images were coincident the
difficulty remained that the images could not be of the same size.
Huygens's conclusion that in practice the weaker image would
not be distinguished by the eye if the images were coincident
led him to renewed experiments but the work does not seem to
have progressed very far.

To these last years belong the remainder of his work on his
Dioptrica, a paper on harmonics, his continued studies in mathe-
matics partly in collaboration with David Gregory, who visited

94 THE LIFE OF CHRISTIAN HUYGENS

him in 1693 and the clarification of certain points in his atomic
theory. Huygens was a strong supporter of the atomic doctrine
which was later to provide the basis of a scientific development
of Chemistry, but he is not usually numbered among those who
developed the atomic theory simply because his researches led
him in the direction of Physics. He continued to oppose Newton
over the doctrine of universal gravitation and added to this his
difference from the English scientist over the corpuscular theory
of light then gaining wider acceptance. The extreme rarity and
speed of the corpuscles and the absence of any obvious way of
explaining colour on this hypothesis were Huygens's main objec-
tions to the theory which was for a century to overshadow his
own elegant treatment of the propagation of light. Huygens
however, was not inclined to seek controversy. In 1694 he had
occasion to correct Renau, an engineer to Louis XIV, over his
connotation of force, a matter which was too important to over-
look even in his declining health.

Huygens himself recognized that the illness which had
dogged him since his return from Paris had become much more
threatening. In March 1695, Huygens felt it necessary to call his
lawyer and make the final corrections to his will. The following
month he became worse and from then until July, pain and
sleeplessness spared him hardly at all. He lived in dread of
losing his reason and his days were filled with a deep despair.
Constantin visited him for a few days at the end of May. Neither
he nor the Calvinist pastor who was at last brought against
Huygens's wishes could give him comfort. Against the doctrines
of personal immortality and the exhortations of the Reformed
Church he maintained an obstinate scepticism. Huygens died
facing the problem of individual and personal immortality as " a
problematical question/' his attitude in this contrasting with that
of other seventeenth century scientists, especially Boyle, Pascal
and Newton.

Nevertheless, when the end came during the afternoon of
July gth, the failing spirit, weakened by suffering, may have
found some serenity. The author of Cosmotheoros, it has been
well remarked, revealed himself, in the face of problems less
abstract than those of mathematics and mechanics, as sensitive
to the deep realities of life and the profound aspirations of the
human spirit. But he could not forget that all experience may be
subjected to the scrutiny of a dispassionate mind, and dispa*

THE LIFE OF CHRISTIAN HUYGENS 95

sionate and remote Huygens always seems to have been. He
lacked perhaps, a sense of the mystical and preferred to look on
life and death with the vision of one to whom all things
appeared capable of an ultimate rational explanation. For him
as for Mersenne, the technique and method of the sciences were
exalted because they freed men's minds from error. But for
Huygens there was no " two-fold truth". Truth for faith and
truth for reason were, for him, one.

The professional and serious interests of Huygens are the
ones which are uppermost in his correspondence. Nevertheless,
it would be a mistake to consider him as always having been
nothing but a patient researcher. He was a man of wide culture
and of acquaintance throughout Europe. The poetry and music
of his own country, with which he was well conversant, were of
a high order; its painting reached the summits marked by the
names of Rembrandt, Franz Hals, Vermeer. In Paris, Huygens
used frequently to visit noted musicians, among them the
harpsichordist Chambonniere. He himself played the harpsi-
chord. Nor was he averse to feminine society. One reads of his
meetings with Marie Perriquet at the country house of the
academist Conrart, with fashionable women at the lectures given
by Rohault. Still more interesting is the fact that he was an
occasional visitor to the salon of Madelleine de Scudery.
Marianne Petit, daughter of one of Louis XIV's engineers, seems
to have had especial charms for Huygens but she never married
and their separation was due to her withdrawal from society
when she entered a religious order. There were even scandals
about Huygens during the war of 1672-8, for he paid frequent
visits to Madame Buat, a grand-daughter of the Dutch poet
Cats, but not possessed of the highest reputation for virtue.
There were, too, some distant cousins whom he visited in Paris,
and for the eldest of these there is no doubt he felt considerable
attraction.

Huygens moved in an elegant and leisured society during his
years in Paris and there are echoes of days at Viry, where the
Perraults had a fine country house, as well as faint glimpses of an
elegant Huygens visiting occasionally the salons of the city.
The contrast of this varied life with the quiet of his days near
the little village of Voorburg must have been marked. But in
fact great changes were taking place and Catholic intolerance in
France was undoing much that Colbert had been at pains to

96 THE LIFE OF CHRISTIAN HUYGENS

build. After 1685, lt was impossible for Huygens to think of
returning to Paris. Holland remained what she had been early in
the century, a refuge for free spirits, a country where new
thoughts might still be brought into the world. But Huygens
was past the days of his vigour and little that is new can be
attributed to his later years. As an exceptionally placed observer
he had seen a good deal, moving as he had done, in the diplo-
matic and learned worlds of his time. The political absolutism
of the age must have contrasted strongly, one would have
thought, with the new freedom of speculation which survived all
repressive measures. But Huygens does not appear to have
remarked on this. The doctrines of expediency, sovereign right,
vital interest and the like have survived, and freedom of thought
requires to be continually defended. That much is clear. But
Huygens belonged really to the world of abstract thought
and shrank from contact with political affairs. In his own world
he was as Leibnitz and John Bernoulli agreed, " incomparable ".
His loss, Leibnitz averred, was "inestimable" for he had equalled
the achievements of Galileo and Descartes; helped by what they
had done he had surpassed their discoveries: " In a word he
was one of the greatest ornaments of this time."

PART II
HUYGENS'S SCIENTIFIC WORK

The State of Science in the First Half of the Seventeenth

Century

THE task of this chapter is not so much to describe actual
achievements of the men of science as to explain how at this
period some of them looked at Nature, and to indicate what
particular problems they felt called upon to solve. Such a ques-
tion bristles with difficulties. It may be that it is of the type
which beckons on only the foolhardy. However that may be, it
certainly must be admitted that the views expressed are neces-
sarily of rather a personal character; they will invite, on some
points at least, as much dissension as agreement.

Very probably there is a good deal to be said for the view
that steady progress in the physical sciences had to wait until
the seventeenth century because of the unsuitable manner of
analyzing events which came from the study of Aristotle. Aris-
totle's interest, undoubtedly, lay rather in logical relations and
his notion of change fitted in with the conception of develop-
ment of form, of growth of living organisms, far better than it
did with the phenomena of the inanimate world. Rejecting the
notion of sheer physical determinism somewhat as Plato had
done, he directed all attention to the purpose served by events
and hardly any to the conditions which were invariable ante-
cedents. The question why things happened as they did was
always tempting and appealed to men as an outlet for tempera-
ment; it was so much less restrained and impersonal than the
question how things are caused, and so long as men's minds
were bemused with an elaborate hierarchical scheme the latter
scarcely seemed to matter. For Aristotle held that change is
purely a terrestrial affair, at least it must be limited to the sub*
c 97

98 HUYGENS'S SCIENTIFIC WORK

lunary sphere, and within that sphere there were just a few
natural motions through which the different elements could
attain their proper level. The natural motion of " earth " down-
wards, or of " fire " upwards, were thus matters which invited
no discussion. For " unnatural " motion, such as that of a
lump of earth horizontally, a constant push must be applied. For
motion in a circle, on the other hand, as in the case of the
apparent motions of the heavenly bodies, no acceleration to-
wards the centre was ever dreamed of; Aristotle had nothing
but reverence for this form of motion for it was, he taught, the
perfect form, and proper only to the celestial bodies.

Galileo's breaking away from scholastic doctrines must have
come after a period of perplexity, for he no doubt tried out cur-
rent explanations for what they were worth. From Roger Bacon
to the Accademia del Cimento is a long distance in time and
there was a considerable change in outlook. The difference was
that the Accademia existed to learn about the world through ex-
periment and to cast its explanations in a form which was no
longer subservient to religion. But in such an intellectual climate
as existed at the very beginning of the seventeenth century the
only safe plan was to concentrate on the description of motion in
geometrical terms and this, of course, Kepler was fortunate
enough to do. Kepler stood little danger of the condemnation
meted out to Galileo in spite of his almost equally great fame,
because his work could be regarded as a presentation of Aristotle's
formal cause in an elegant mathematical form. Mathematics, in-
deed, remained respectable, supported still by the prestige of
Plato, and in the sixteenth century there was in some of the Italian
centres of learning a renewed interest in the teachings of Pytha-
goras. From Pythagoras, Whitehead has remarked, " mathema-
tics and mathematical physics took their rise. He discovered the
importance of dealing with abstractions; and in particular
directed attention to number as characterizing the periodicities
of notes of music. The importance of the abstract idea of periodi-
city was thus present at the very beginning both of mathematics
and of European philosophy." For Kepler " the cause of the
observed effects is the mathematical beauty and harmony dis-
coverable in them ".

The history of this mathematical treatment of phenomena
is usually somewhat briefly dismissed, as if it all resulted from the
insight of Galileo and Newton and all the rest were fools. But

SCIENCE IN THE EARLY SEVENTEENTH CENTURY 99

Copernicus, it should be remembered, showed a fundamentally
similar attitude much earlier, although neither he nor Kepler con-
cerned themselves with the physics of their problems. By 1543,
the date of his De Revolutionibus Orbium Codestiwn, he had
analyzed linear simple harmonic motion as the product of two
combined circular motions. This is the first record of a study
of such a form of motion which Copernicus himself described
as pendentibus similes after the manner of suspended bodies.
But, characteristically enough, circular motions were still
thought of as raising no particular problems. Benedetti, how-
ever, in his Disputationes of 1585, showed an advance in that he
adduced linear simple harmonic motion in controverting some
of Aristotle's ideas about motion. Then, as against Aristotle's
teaching that bodies fall towards the centre of the earth because
that is their " natural " place, he asserted, though he did not
prove, that if a weight were dropped through a hole passing
through the centre of the earth, it would by no means come to
a stop but would oscillate backwards and forwards in the manner
of a pendulum bob. This interest in accelerated motion marked
the beginning of the criticism of Aristotle which was to grow
so much in intensity during the next hundred years. It may
indeed be said that modern science begins with the study of
dynamics and in particular with perplexities about simple har-
monic motion.

Galileo, in his Dialogue concerning the Two Chief Systems
of the World (1632), made considerable use of the simple pendu-
lum in his study of naturally accelerated motion. He restated
Benedetti's view of the subsequent motion of a body dropped
through a hole passing through the centre of the earth. Dis-
satisfied with the formal cause which set such limitations to Kep
ler's work, he looked, in the case of accelerated motion, for what
Newton later called an external impressed force. He even had
some notion of the necessity of postulating action and reaction,
for in writing of the simple pendulum he perceived that the
thread, in so far as it possesses mass, must "hinder" the motion of
the bob and that there was a problem the problem of the com-
pound pendulum which remained to be solved. His study of the
simple pendulum shows, however, a necessary amount of idealiza-
tion, without which a relation between the period and the length
would have been excessively hard to find. Galileo's work, in fact,
marks the first successful union of experiment and mathematics,

100 HUYGENS'S SCIENTIFIC WORK

for he brought together the criteria of the simplicity of natural
laws and their accordance with experiment. The problems
he bequeathed were as important as those he solved: the
concepts of mass and momentum, and of force and work, and the
law now known as Newton's Third Law were all apprehended in
only a somewhat confused way at Galileo's death. Mersenne's
Hannome Universelle (1636) is not regarded as being of such
moment as the writings of Galileo, to which Mersenne was in fact
greatly indebted. He deserves credit, however, for being the first
man in Europe to perceive Galileo's genius and to promulgate
his ideas.

So far all that has been said has been concerned with the
bright and positive side of the picture of the early seventeenth
century, but there was a vast difference between the conditions
which then ruled and those of today. Today we are accustomed
to an intellectual demarcation of Science and Philosophy and
to almost equally sharp boundaries between the various sciences.
Then there was a chaotic muddle of " philosophies " and no tacit
agreement on the fundamental concepts through which phenom-
ena are ultimately explained. If we single out the great
names it is possible to see the seventeenth century as a period of
amazing clarification, but this it must rarely have appeared to
be to the contemporary men of science. The alchemists, for
example, were engaged in the most obscurantist practices, and if
Science as a whole had depended on their progress in their " art "
the period would have to be written off as a failure. The works
of Paracelsus, Basil Valentine and Raymond Lully which con-
tinued to circulate, and the extensive forgotten literature pro-
duced by a host of enthusiastic and credulous amateurs, take us
back to the worst periods of superstition and it is simply not true
that the dawn of the seventeenth century brought a universal
belief in the operation of physical law and a discredit of
magic and mystic agents. There was this one brilliant triumph
of mechanics and theoretical astronomy, but so acute a thinker
as Descartes missed the true answer to the sceptical questions he
set himself. By falling back on an intuitional method he was no
doubt able to shorten his estimate of the labour required in this
way to produce a complete science of inanimate nature, but this
was in essence the same mistake as Aristotle's. Only those with
a weakness for verbal and logical classification could follow Des-
caries willingly through his opening premises and voyage thence

SCIENCE IN THE EARLY SEVENTEENTH CENTURY IOI

through the world he constructed. There was, of course, the
enormous difference that Descartes 's universe was supposed to be
governed throughout by physical laws, even though these hap-
pen not to be the ones which actually operate. But the idea was
there, and the conception was one of the universal mechanical
operation of law.

By comparison, the atomists who followed the teaching of
Gassendi seem to be intellectually children, and the atomic doc-
trine, by 1650, was not noticeably an improvement on the original
version of Lucretius. Its adoption by Hobbes, a friend of Gas-
sendi, and the most uncompromising materialist of them all,
was a misfortune, for Hobbes's reputation with the men of science
became such that he had only to support a theory and it was
damned. Much the most important contribution to atomic
theory came in fact from Boyle, and we may see in certain brief
essays in this style an embryonic form of the kinetic theory.

But the existence of so many principles of explanation, drawn
from Aristotle, Pythagoras, Lucretius, Descartes and Galileo, and
the inability of some men of science to distinguish what was
fact from what was mere hypothesis, all this made for unneces-
sary obscurity. The discipline which characterized mathematics
and astronomy had somehow to be introduced into a field which
too easily rioted into thickets of explanatory hypothesis. Refer-
ence to authority became prohibited; an anti-intellectualist atti-
tude had to be encouraged and we find that William Gilbert, for
example, denounced the " vast ocean of books " produced by
writers whose explanations exhibited merely verbal ingenuity.
" Neither Greek arguments nor Greek words can assist in finding
truth/' he added. True, Francis Bacon wrote in a similar vein,
but to a less extent than Gilbert was he able to explain the
method which should be employed; indeed, he hindered progress
by his persuasive suggestion that there was one particular method,
and especially when he argued for classification of facts and
omitted to consider measurement. Nevertheless, the men of
science succeeded as much by practical ingenuity as by abstract
reasoning in reducing their problems to some sort of order, and
by the time the Paris and London societies came into existence
some confidence was felt that natural philosophy could and
should contribute in no small degree to the material welfare of
men. That natural philosophy might profoundly alter men's be-
liefs about the world was not then thought likely, and it seems

IO2 HUYGENS'S SCIENTIFIC WORK

that this modesty on the more intellectual aspect was genuine.
The Cartesian scheme went as far as the educated man could
desire in showing how the claims of Biblical cosmology and
natural philosophy might be reconciled. And up to 1670 it does
not appear that fundamental researches were planned with any
expectation of a new conception of the universe, doubtful though
Huygens and others might be of the validity of some of Des-
cartes's theories when examined in detail. Quite a large propor-
tion of the scientific work going on in the i66o's was concerned
with practical things : mining, navigation, military science, tex-
tiles and so on. Following Bacon, the men of science seemed
to believe that "the real and legitimate good of the sciences
is the endowment of human life with new inventions and
order ".

There were good reasons for this practical bias, for many of
the techniques of civilization had advanced relatively little over
a period of centuries. With the expansion of commerce, for
example, navigation particularly remained difficult owing to
the lack of a reliable and practicable means of determining long-
itude at sea; the geographers and cartographers had in this
respect also gained little since the twelfth century. Hence it was
that when Galileo discovered Jupiter's moons quite as much
interest was attached to their use as a source of standard time as
to their cosmological significance, representing, as they appeared
to do, a solar system in parvo. Not until 1668, however, were
ephemerides for the moons published; then their publication by
Cassini gained for the astronomer the recognition of an invita-
tion to Paris. The secret of longitude determination could clearly
mean a great deal for the national income, for ships had to
struggle on their voyages to the East Indies by means of dead
reckoning and such checks of longitude as could be made by
observation of landmarks. Bond tried to use magnetic isogonals
as an answer to the problem and his book The Longitude
Found (1676) was considered to be important by Halley, though
nothing came of the method in practice.

Astronomy remained the key to many of the problems which
were felt to be urgent during the first half of the century and
it was the science which was most advanced through the work of
Tycbo Brahe, Kepler, Galileo and others. The invention of
logarithms by Napier in 1614 was timely for, in Laplace's words,
it doubled the lifetime of astronomers though Napier's labours

SCIENCE IN THE EARLY SEVENTEENTH CENTURY 103

were greatly inspired by the aim of proving the Pope to be Anti-
christ. So soon as money could be obtained, the French decided,
an observatory should be constructed at Paris. It must be remem-
bered that as yet a convincing decision between the Ptolemaic,
Tychonian, and Copernican systems had still to be given; even
though the first was no longer seriously supported by men of
science, it shared with Tycho Brahe's system the advantage that
no observation of stellar parallax had ever been made. The sup-
porters of Copernicus could attribute the absence of parallax
only to the smallness of the effect and this depended on the
dimensions of the solar system being very great.

When, accordingly, as a result of the energy of Auzout,
Richer, Picard and others, an expedition was sent to Cayenne in
order to make simultaneous observations of Mars there and at
Paris, the size of the solar system was the primary question in
view. Richer at Cayenne and Cassini at Paris took simultaneous
measurements of the altitude of Mars and in 1677 the planet
was at its nearest to the earth. Knowing the base d of the triangle
in the figure, and the base angles % l and 9 2 , the distance of the
planet was easily computed.
By a straightforward appli-
cation of Kepler's laws of
planetary motion all the
required distances followed
from a single determination
of the distance of the two
planets. For the sidereal
times of the orbits were
known and these were re-
lated with the mean dia-
meters. Kepler's brilliant
work on the orbit of Mars
and this, triumph of 1672
thus alike rested on the
patient labour of Tycho Brahe and the whole was a sermon
to the men of science on the power of quantitative
mathematical relations. The conclusion that the sun's mean
distance from the earth was 87,000,000 miles was of the right
order. As J. W. Olmstead remarks in I sis (1942), writing of the
expedition, " The great consequence of the expedition for con-
temporaries was, of course, the revelation of the tremendous

FIG.

IO4 HUYGENS'S SCIENTIFIC WORK

dimensions of the solar system, as well as the prodigious size of
the sun and some of the planets The disclosure, with some cer-
tainty, of the gigantic distances and masses involved was, for the
general public, almost overpowering." The old Aristotelian con-
ceptions of the universe might linger on in some minds but
these were shattering blows and it was inevitable that educated
men of the period should accommodate the increasing know-
ledge of Nature, up to 1672, with the only other plausible " sys-
tem " in existence : that of Descartes.

It may be true that, as Professor Andrade has remarked in
Nature (1942), " The Cartesian scheme was easy, pictorial, gen-
eral : the Newtonian difficult, mathematical, precise. " In such
an opposition it may appear that Descartes's ideas were woolly
and essentially unscientific. Nevertheless, it does not do to under-
estimate the amount of preparation which preceded the New-
tonian synthesis or to despise investigations of men whose minds
were, at least tentatively, inclined towards Descartes's theories.
However deplorably qualitative his theorizing, Descartes inspired
a generation at least with a vision of Nature as a sphere of
universal law embracing planetary motion, rainbows, the pro-
perties of magnets and of lenses, all that could be explained
by reference to underlying mechanisms. There was no
limitation of the field of scientific investigations, and, more-
over, the scientific explanations Descartes looked for were to
possess an admirable economy of principles, the subtly matter
and ether being trusted to account for a variety of phenomena.
It was his desire for a pictorial mechanism which was
the greatest weakness of Cartesianism, and if it had proved
possible to reconcile his ideas with measurement and so formu-
late mathematical relationships, there is no doubt that Descartes
would have done this. But he was willing to sacrifice the world
of observation if need be, and he did not see that there was
a radical difference between Galileo's idealization of his
experimental studies and his own bold embarking on pure
supposition.

All this became clear enough to Huygens in the course of his
life; but the problems he took up can frequently be understood
best by reference to Descartes's essays on the same subjects : the
laws of impact, centrifugal force, the centre of oscillation, the
behaviour of lenses, the nature of light and the cause of gravity,
these are all topics discussed by Descartes and in a manner which

SCIENCE IN THE EARLY SEVENTEENTH CENTURY 105

serves to introduce Huygens's own exact investigations. Professor
Andrade makes an excellent remark when he says : "What New-
ton ignores is what Aristotle and Descartes tried to start with ",
and his summary of Newton's achievement cannot be ques-
tioned : " If we are asked to state in *a sentence what was the
main effect of Newton's work on the thought of his time, I think
that the answer must be that it was to establish the power and
universality of the methods of quantitative science." But ever
since the time of D'Alembert's "Preliminary Discourse " to the
Encyclopedic of 1751 there has been a tendency to admire New-
ton's magnificent work by depreciating the earlier work of the
century, and especially everything which could be labelled Car-
tesian. It is important to realize what useful work was in fact
done by the men of this period, and when we come to a con-
sideration of Huygens's work much allowance has to be made
for the late appearance of his works as publications. Otherwise
the effect outside the Academic might have been far greater.
" The evidence shows," wrote Professor Harcourt Brown, 1 " that
Huygens took up the best elements of Cartesianism, adding to it
along certain lines, rejecting some of it in other directions, and
that he followed whatever seetned most fruitful of the sugges-
tions received from parallel studies in England, Italy and else-
where. He counts as one of a group whose efforts prepare the
modern age of science, ushered in by way of the encyclopaedism
of the eighteenth century, whose urbane and moderate scepti-
cism is a definite premonition of the century of Voltaire."

It must be said that where Huygens's views most foreshadow
the eighteenth century was in his acceptance of complete physi-
cal determinism, even though he chose to cast this in rather rigid
mechanistic form. This chapter may, therefore, end with a brief
discussion of the mechanistic hypothesis of which Huygens was
so strong an exponent. The influence of Gassendi may be seen
from time to time, even though Descartes's system left little scope
for the development of Gassendi's atomic doctrines and the two
philosophers were in some senses regarded as being in direct
opposition. But Gassendi seems to have strengthened in Huy-
gens a leaning towards the thoroughly mechanistic view of
Nature. Kepler had supposed that magnetic attraction resulted
from a " mutual affection "; Gassendi considered it was a purely
physical force. For him light was a material substance; there
* Annals of Science, 1936.

106 HUYGENS's SCIENTIFIC WORK

were indeed atoms of light and heat and for the propagation of
sound an example of a doctrine carried to extremes. Such views
were opposed to Descartes's conceptions, for the latter held that
space or " extension " is not infinitely divisible. Both considered
that the universe should be regarded as a machine, although in
the last resort Gassendi seems not to have thought it regulated
itself. It was a scientific form of this teaching which came out
of Huygens's mechanical studies, although he himself preferred
to observe a separation between science and philosophy. Cer-
tainly, so far as they went, Huygens's mechanical explanations
showed several radical differences from those associated with
the name of Newton. In particular, Huygens found great diffi-
culty in making universal gravitation a part of his scheme, pre-
ferring rather to regard matter as itself quite inert and incapable
of any action except through collision with other matter. This
difficulty meant in practice that he remained a Cartesian. In
1646 Descartes had ridiculed the " absurd belief " of Roberval
that there could be mutual gravitation between lumps of matter.
It would mean, he said, that a particle of matter was possessed
of a soul and endowed with consciousness, so that it could know
what happened across space and could in some occult manner
exert its own influence there. For a Cartesian the really urgent
problem was rather to discover the laws governing the transfer
of momentum on impact and this, as we have seen, was one of
Huygens's earliest studies.

It seems quite certain, therefore, that some of Huygens's
rather curious conceptions come from Descartes. To the reader
of Descartes's Principia Philosophise, for example, there would
appear to be no incongruity in treating light and mechanics as
related subjects. Descartes, and following him, Huygens,
thought to find the explanation of gravity and light in the pro-
perties of media which would enable bodies to act on each other
across intervening space. Huygens perhaps did for a time
seek to bring in that mathematical #nd scientific form of Des-
cartes's theories which Paul Mouy supposed that he actually
created *. But in fact, while he was at the very beginning of his
work, Huygens saw that there was exceedingly little truth in
many of Descartes's scientific " principles ". On the other hand,
he could not adopt Newton's attitude, which left the question of
pictorial mechanism out of consideration to a very large extent.
1 Le Ddveloppement de la Physique Caftdsienne (1934).

SCIENCE IN THE EARLY SEVENTEENTH CENTURY 107

As has been mentioned, Huygens considered that the conception
of a gravitational force, innate and inherent in matter, appeared
to be absurd and one which " takes us very far from the prin-
ciples of Mathematics or Mechanics ". As an enthusiast for a
kinetic theory of all change he could allow matter no properties
save that of inertia. It has to be remembered that he considered
he had obtained direct evidence for the existence of a subtle mat-
ter. But for Cartesianism as a philosophic system he had little
patience and he accordingly had several clashes with some of the
Jesuits and doctrinaire Cartesians. When Kircher adopted the
cosmology of Tycho Brahe, Huygens criticized him for what
appeared to be nothing but timidity. " We others ", he asserted,
are " without fear ".

In a sense then, Huygens's support of some of Descartes's
ideas up to so late a date in the seventeenth century must be
described as an experiment in failure. Very probably it was an
experiment that had to be made; its failure showed clearly the
great limitations of the extension of the notions of sense in the
field of theory. It may even have been the feeling of dissatis-
faction with the position men like Huygens were reaching that
drove Newton to make his great renovation of natural
philosophy. The progress of scientific explanation may then be
seen as a process of leaving out of redundant elements until one
arrives at the really successful procedure of abstraction. If so,
this was a return to the methods of Galileo.

The practical value of much of Huygens's laboratory work
needs no stressing, nor does his interest in the clock, telescope,
microscope, and other experimental and observational aids need
explanation. The experiments of Galileo and the Florentine
academicians were done with the simplest of apparatus. But new
discoveries, it was realized, often attended the use of new appara-
tus, a fact which was well illustrated by the invention of the
vacuum pump. The rate of progress was for some time in direct
relation with the experimental resources available. Here theory
and practice interlock. It was not possible to improve the tele-
scope, to invent a new eyepiece, without a quantitative study
of the defects of the ordinary lens. Problems of theoretical in-
terest led to the construction of the vacuum pump; experiments
with this instrument led in their turn to new theoretical specula-
tions regarding the nature of air, and, in Huygens's case, the
cause of gravity. One can scarcely estimate the extent to which

108 HUYGENS's SCIENTIFIC WORK

new and improved instruments assisted the great scientific
advances of the seventeenth century. The telescope, microscope,
barometer, vacuum pump, pendulum clock, micrometer eyepiece,
and other instruments might be mentioned here. Without them
rapid progress would have been unthinkable.

77

Work on Collision between Elastic Bodies

The Cartesian mechanism was governed by the principles of
inertia and of the conservation of motion, the laws of impact, the
theory that solidity is derived from a condition of rest, and a
kinetic theory of fluids. Independent of Galileo in most respects,
Descartes expressed the Newtonian First Law quite clearly: by
itself a body continues in its state of rest or of uniform motion
in a straight line; curvilinear motion is constrained. We know
that Galileo also stated this law. Descartes went further than
Galileo, however, for he considered that rotational motion should
generate an outward, centrifugal force. In his vortex theory this
centrifugal force was balanced by the pressure of neighbouring
vortices, also tending to expand. Each vortex had, in feet, a cen-
tre which was comparatively empty of matter and consequently
matter leaving the ecliptic of one vortex was drawn in at the
poles of a neighbouring vortex. Descartes thought to explain
magnetism and sunspots by this means. Two other phenomena
arose in consequence of the vortices : the centripetal attraction
on matter resulting from the emptiness of the central part of
a vortex produced the illusion of gravity, and the centrifugal
pressure of particles of the subtle matter was the origin of light.
Descartes never really answered the criticism that gravity acts
along the radius and not along lines perpendicular to the plane
of rotation and there were other difficulties for his view. In the
first place he did not consider that the rotating subtle matter
penetrated the earth. Yet the weight of an object may be shown
not to vanish beneath the earth's crust. Also Descartes did not
consider weight as proportional to the quantity of matter and he
supposed that liquids, through the greater internal motion of
their panicles, must be inherently lighter than solids. Huygens
saw these weaknesses. Nevertheless, he followed Descartes on the
subject of gravity to a greater extent than in other topics. He
even calculated how fast the subtle matter must move in its
vortex to produce the known effects. There are other parallels,
too, in Huygens's theory of the transmission of light. Neverthe-
less, in the pages which follow it will be seen that Huygens

109

HO HUYGENS'S SCIENTIFIC WORK

remained outside the main stream of Cartesian thought which is
to be traced in the writings of Rnier, Regius, Rohault, Regis,
Fontenelle and Malebranche. On the other hand, he was not in
spirit so deeply opposed to Cartesianism as Leibnitz, whose view
of the Principia Philosophise was that it was " un beau roman
de Physique ".

In this work Descartes supposed that motion could only be
transferred from one body to another by direct or indirect im-
pact. His assumption that the total amount of what he called
the " quantity of motion " in the world is constant was not un-
reasonable : it conceals the germ of the idea of energy which
may be traced in the writings of Stevin, Da Vinci and perhaps
in those of Jordanus Nemorarius. But Descartes made a distinc-
tion between the speed of a body and its " determination " or
direction which led him to treat these as separate entities. In his
third law of motion Descartes stated that if a moving body, tra-
velling in a straight line, meets another body with less " force "
to continue its movement than the second one possesses to resist
it, it will lose its determination without losing its motion. If it
has more " force " it moves the second body and loses as much of
its motion as the second one absorbs. Descartes's seven rules of
impact were all corollaries of this law and so important were
these in his system that Mouy has remarked: " Les regies du
choc sont bien les regies du mouvement et la m^canique carte-
sienne est une theorie de la percussion." It was unfortunate,
therefore, that all the rules were wrong when compared with
experimental results. Descartes appears to have known this and
to have replied in effect: the rules are true for perfectly hard
bodies moving in uacuo. In experimental conditions the bodies
are not perfectly hard and move in a fluid. The parts of this
fluid " corrupt " more or less the bodies immersed in it owing
to the motion of the fluid particles against the particles of the
body.

During the year 1652, when he was only twenty-three, Huy-
gens became convinced of the errors of Descartes's treatment of
motion and impact. Correspondence with Schooten, Gutschoven
and Slusius in the period of 1652-7 shows that he was occupied
in substituting new principles for those referred to above. The
fact that Huygens's work 1 was not published as a whole in his
lifetime is certainly not to be attributed to his reverence for Des-
1 It was entitled De Motu ex Percussione.

COLLISION BETWEEN ELASTIC BODIES

I II

cartes but to his own difficulties over further dynamical prob-
lems. Actually the work was completed before 1656, for in that
year Huygens undertook the recasting of his material in classical
form. Some twelve years, therefore, elapsed between Huygens 's
completion of the treatise and the communication of his results
to contemporary men of science (see p. 64).

Huygens at first limited himself to the study of direct col-
lision between two equal bodies. He began with two hypotheses.
The first was Galileo's principle of inertia, also stated by Des-
cartes; the second stated a principle of symmetry. The latter
was that if two exactly similar hard bodies collide with equal
speeds in the same straight line they are reflected back with their
speeds unchanged. He uttered a warning that this was true in
respect of the system of co-ordinates used in specifying the posi-
tion of the bodies at any instant and that the system might be in
a state of uniform motion. Velocities relative to the system of
co-ordinates or reference frame he saw must be treated as vectors
as forces were already treated. His problems were of course
concerned with momenta, but this fact is a little obscured by his
geometrical method of treatment in which the velocities alone

Q-

C

O

FIG. 2

are represented. The bodies were supposed to be smooth spheres
of a perfectly elastic material. They were imagined as suspended
by vertical strings held by a man standing in a boat. The
boat or system of co-ordinates could be given a uniform
velocity to the right or to the left so that the velocities of the
spheres, as judged by a stationary observer on the bank, were
ijn effect transposed to a different system or reference frame.

Thus, if the spheres have velocities v and -v (Fig. 2) and the
reference frame moves with a velocity -v the first sphere is made

112 HUYGENS'S SCIENTIFIC WORK

to appear stationary and the second to approach with a velocity
of -2V. After impact, with respect to the frame, the velocities arc
-v and v, but with respect to the stationary observer they appear
to be -av and o. Thus, when one body is stationary and is struck
by a moving body of equal mass, the momentum is transferred.
In the case where the body A has a velocity v l and JB has a velo-
city -v a where v x > v 2 , Huygens supposed the reference frame to

be given a velocity - so that, to the stationary observer, the

velocities appear to be equal and opposite. It followed that the
real velocities are interchanged.

To progress in problems where the masses are unequal Huy-
gens found it necessary to assume as self-evident that when a
larger body meets a smaller one which is at rest, the smaller
one is moved and the larger one loses some of its motion. Here
he was engaged in clearing away some of Descartes's errors, some
of which arose through his failure to treat velocity consistently
as a vector quantity. In March 1669, in the Journal des Savants,
Huygens gave a correct statement of the law of conservation of
momentum in the form: " There always remains the same
quantity of motion towards the same side after the quantity of
contrary motion has been subtracted." Another form of
law is given in Rule 7 of the DeMotu: "... the common
centre of gravity of the bodies advances always equally towards
the same side in a straight line before and after impact."
That is, the total momentum along the line of centres
is unchanged. It followed directly from this that the relative
speeds of approach and separation are equal.

Much the most interesting part of the De Motu is found in the
proposition on the collision of bodies when the velocities are
inversely proportional to the masses. 1 " When two bodies, whose
speeds are inversely proportional to their masses, meet from
opposite sides, each rebounds with the same speed with which
it approached."

Here Huygens employed the important principle, referred
to in Part I, that the centre of gravity of a system of bodies
cannot ascend through any motion of the bodies under gravity.
His proof consisted in showing that if the resultant speeds were
other than equal and opposite to the original speeds, the centre

1 Proposition 8. Huygens did not make a distinction between mass and
weight in this work. The first use of a distinct term for mass comes in his
work on centrifugal force (see p. 117).

COLLISION BETWEEN ELASTIC BODIES 113

of gravity of the bodies could be raised above its original height.
This, following an idea found in the writings of Stevin and
Torricelli, he assumed to be impossible, since it involved a
possibility of perpetuum mobile. In the figure the mass m A pos-
sesses the velocity AC, while the mass W B possesses the velocity
BC. These are related:

tw A = BC

m B AC

FIG. 3

After impact, according to the theorem, the velocity of m A will
be represented by CA and that of ra B by CB. Using a reductio
id absurdum, Huygens first assumed that the acquired velocities
ivere CD and CE, the only necessary restriction here being that
AC + BC = CD + CE (ignoring signs). Supposing that the
initial velocities were acquired by vertical descent under gravity,
Huygens took the required heights to be HA and KB respec-
tively. These were related through Galileo's equation v* = 2gs

[Fig- 3)-

HA _ AC'

1 : ~KB ~ C&

Starting with the velocities CD and CE the heights attained
would be AL and BM such that

AL _ CD 3

~HA ~ AC*'

BM _ CE*
and KB ~ OP'

H

114 HUYGENS'S SCIENTIFIC WORK

When first elevated ra A and w B have their common centre of
gravity at N. After being brought to the positions L and M , the
centre of gravity is moved to O for

m^.AC = ra B .CB
HN AC LO

NK CB OM
The remainder of the proof then consists of a simple demonstra-
tion that O is higher than N not only for this case but also for
the case when ra A rebounds with more speed than before and for
the possibility that it is brought to rest.

It will be noticed that the idea of converting a horizontal
velocity into an ascent involves first the relation v* = 2 gh and
second the idea of energy conserved, i.e., mv 2 = 2 mgh. No
doubt it was the use of this relation which enabled Huygens to
solve problems on the ballistic pendulum when in England in
1661. It marks a distinct advance on Descartes 's approach to the
law of conservation of energy, since the latter had no use for any
form of potential energy. This piece of work is accordingly of
some importance in the history of the concept of energy.

After this point Huygens was able to give the solution of the
most general case of impact between perfectly elastic bodies mov-
ing along the line of centres. The masses being in general
unequal, Huygens employed his original device and gave the
reference frame a velocity such that the velocities of the bodies
bore an apparent relation inversely proportional to the masses.
His ^geometrical method prevented him from expressing the
result in a generalized form. The most general law embodied
in the work on impact is the law of conservation of kinetic energy
for linear motion in the horizontal plane. This is stated in a
proposition showing that ra A ^ 2 A -f m B z; 2 B = constant, both be-
fore and after a collision. The difficulty of dealing with vector
quantities is overcome by squaring the magnitudes. How Huy-
gens hit on the quantity mv 2 is not quite clear, but his use of it
appears to date from about 1652 and he considered this, the
efficacy or vis viva of a moving body, to be a more fundamental
quantity than its quantity of motion or, as we should now say,
its momentum. Leibnitz adopted the same point of view, but
neither he nor Huygens saw how the vis viva was related to the
quantity of motion.

The remainder of Huygens's De Motu Corporum ex Percus*

COLLISION BETWEEN ELASTIC BODIES 115

siane deals with the effect of placing hard bodies between the
two extreme bodies in motion so that the impact is transmitted.
In the simplest case a hard sphere is brought to rest on meeting
the end sphere in a row and the sphere at the other
extreme end is set in motion with the velocity originally
possessed by the first (Fig. 4.) Huygens experimented with glass
balls suspended like pendulum bobs or rolling in a horizontal

FIG. 4

groove. It is well known that he found in the study suggestions
for a theory of the propagation of light. He showed in the present
connection that a single interposed mass must be the mean pro-
portional between the two extreme masses for the velocity
attained by a third mass to be a maximum. Thus, at first, m x
has a velocity t> A while V B o and v c o. It may be shown that if

B = */^*"~C>

then v c is a maximum for any given values of ra A and ra c . The
proof is straightforward, although tedious by Huygens's method.
As has been mentioned, others besides Huygens brought out
work on the subject of impact. Wallis, in November 1668, read
a paper to the Royal Society in which he also employed the Car-
tesian quantity of motion but dealt with inelastic bodies. In
167 1 he published results for elastic impact, including the formula

mv m.v, .

u for inelastic bodies,

m -f mi

where v and v l are the initial velocities and u the common
velocity after collision. In December 1668 Wren also gave some
empirical rules which resembled some of Huygens's results,
although arrived at independently.

For a general theory of impact, clearly systematizing con-
temporary work, one has to turn to Newton's Principia, where the
subject is summarized in the introductory Scholium to Book
I. After the three laws of motion Newton gave a correct state-
ment of the law of conservation of momentum and stated as
a corollary Huygens's " law " that the common centre of gravity
of two or more bodies is unaffected by the actions of the bodies

Il6 HUYGENS'S SCIENTIFIC WORK

among themselves. So slowly did the Newtonian system displace
the Cartesian, however, that in the English edition of Rohault's
System of Natural Philosophy, which came out in 1723, Des-
cartes's original treatment of motion and impact is closely fol-
lowed and we read, for example, on page 78 : " When a Body
moves any particular way, the Disposition that it has to move
that way rather than any other is what we call its Determina-
tion. 1 ' Gravesande's Mathematical Elements of Natural Philoso-
sophy (1721) was more up to date and contained a summary of
Huygens's work on impact without, however, an acknowledge-
ment of the source. This work was dedicated to Newton and is
an interesting guide to the scientific heritage of the seventeenth
century as it was passed on to the eighteenth century reader.

Centrifugal Farce

The fact that Huygens early turned his attention to the
study of circular motion is only another example of the influ-
ence of Descartes. In his Principia Descartes attempted to
analyze the motion of a stone placed in a sling. He saw that
although the stone tended to continue its motion along the tan-
gent, this was prevented by the tension in the thread. The prob-
lem was to find the magnitude of this tension and this Descartes
failed to do. Huygens completed his study of circular motion
about 1659 and published the more important theorems at the
end of his Horologium Oscillatorium. The treatise, De Vi Centri-
fuga, containing the proofs of these theorems and other mat-
ter, was published posthumously in 1703. By this time others,
including Keill, Savilian Professor of Astronomy at Oxford, had
become impatient for the proofs of the theorems and had sup-
plied many of them. Newton arrived at the fundamental
formula for the acceleration of a particle describing a circle
independently of Huygens.

Huygens began his treatise with a statement of Galileo's con-
clusions concerning descent under gravity. " Gravity," he said,
" is the tendency to fall " and the tension in the thread from
which a heavy body is suspended results from this tendency. A
Constant acceleration must mean that " the spaces traversed in
different times by bodies starting from rest are to each other as
the^quares of the times ". The tension considered in the case of
the sling is therefore a real
force, as real as a force which
produces an acceleration in
the direction of the existing
motion. Indeed, the measure
of the force is supplied by the
acceleration and, he should
have added, the mass moved.
There is then an acceleration
towards the centre in the case
of circular motion. FIG.

117

Il8 HUYGENS'S SCIENTIFIC WORK

In the figure (Fig. 5), Huygens considered the successive
points reached by a particle liberated at some point of its circular
path and free to travel along the tangent.

These, at successive equal intervals of time, would be K, L, N
in the figure, the distances BK, BL, BN being equal to the
lengths of the arcs BE, BF, BM respectively. If the intervals of
time are small, BK, BL, BN approximate to BC, BD, BS respec-
tively; C, D, S being points on the tangent where this is cut
by the radii through E, F, and M. EC, FD, and MS are then the
distances separating the two paths at successive instants. With-
out explaining his reasoning further Huygens then stated that
these distances approach as a series, the series of squares i, 4, 9,
1 6, etc., and thus resemble the successive distances fallen under
gravity. The argument is somewhat reminiscent of Galileo's
study of a projectile, which, starting horizontally, under the
acceleration of gravity describes a parabola (Discourses on Two
New Sciences, Fourth Day), but he may have got at the result as
we should now by considering a parallelogram of velocities 1 . As
a mathematician he was rather more attracted by the problem of
the nature of the curves EK, FL, and MN, which he saw were the
evolutes traced out by the movement of a stretched thread
wrapped along the arc BE, BF, BM, etc., but he reverted to the
physical aspect of the subject. The distances traversed (EC, FD,
MS) in successive instants being approximately in the ratio of a
series of squares, he wrote, " this tendency of which we have
spoken is absolutely similar to that with which heavy bodies
suspended by a thread try to descend. Whence we concluded
also that the centrifugal forces of unequal bodies moved around
equal circumferences with the same speed are among themselves
as the weights or solid quantities " inter se sicut mobilium
gravites, seu quantitates solides. Professor Crew (The Rise of

a dt
a.dt = v&

8' = (ddJ and <o = v/r

.'. a = v*/r. Under gravity a = v a /25.

CENTRIFUGAL FORCE IIQ

Modern Physics), has remarked that this is probably the earliest
suggestion of a distinction between mass and weight.

From this original beginning some simple conclusions follow
immediately. The centrifugal force varies directly with the
radius of the circle if the angular velocity and mass are constant
(Fig. 7), it varies as the square of the tangential velocity at any
instant (Fig. 8) and inversely as the radius (Fig. 9) :

(a) For equal angular velocities :

D

FIG. 7

FIG. 8

D

FIG. 9

1

DF
EG

BF
CG

BA
CA'

ije. the centrifugal force varies
with the radius (prop. i).

(h v, _ BE

"vl ~ BF

F,
F,

CE
DF

and, by his original relation,

CE __ CB' _ V
DF = DB T = u?'

(c) Equal masses have equal linear
velocities: BD CF - v.

LetCE = v'
v^_CE^_ AC
v "~

BD

But F a

F' :

AB
BA

(prop, i)

and F' AC' ,

F = JR (P r P- 2 )

BA AC 1

AC

AC AB' = AB

120

HUYGENS's SCIENTIFIC WORK

There does not seem to be much necessity to quote the other
propositions on ordinary circular motion since they follow
simply from the above. There are, however, some interesting
studies of the conical pendulum : problems which may have been
a reason for Huygens's taking up the whole subject. As has been
mentioned earlier he made use of the conical pendulum in some
of his clocks.

This section commences with the
statement of two simple theorems or
lemmata on the equilibration of smooth
bodies on inclined planes by means of
tensions in horizontal threads. Thus in
Fig. 10 (lemma i) the weight D must be
to the weight C as the perpendicular
This relation is not proved; Huygens
it is one which is well known. It is

K

FIG. 10

RF is to the base FA
says, justifiably, that

used immediately in an interesting theorem on the revolution
of a body on the inside of a paraboloid of revolution; all revolu-
tions of a body travelling in horizontal circles on the surface of
the paraboloid are completed in equal times irrespective of
the amplitude of the circle. This theorem states the theoretical
conditions to be fulfilled by a conical pendulum used in a clock.
The point about Huygens's proof of his theorem is that, true to
his conception of centrifugal force as equivalent to any other
kind of force, he substituted it for the tension in the thread of
the above lemma. At any point on the paraboloid, then, the cen-
trifugal force must be that which will maintain the body

against the force of gravity along
the tangent. As he showed:

F _ HG = HK
L I mg ~ GF ~ KL

K / But, by a property of the para-

bola, KL is constant (Fig. n).
Thus, for any two positions of a
body on the paraboloid:

G

F

and thus the times will be equal
FIG. 1 1 (proposition i).

In subsequent propositions, Huygens derived all the simple

CENTRIFUGAL FORCE

relations which might be expected: the various factors of vertical
height, length, inclination to the vertical and angular velocity
all being considered separately. Just as he nowhere used the

mv

formula in this form so he did not give the formula

cos for the period of the conical pendulum. The work

o

is consequently somewhat tedious for the modern reader. Near
the end Huygens showed that when the angle with the vertical
made by the thread of a conical pendulum is 2 54' the period is
equal to the time of vertical fall from a height equal to the
length of the pendulum. The work is completed by a consider-
ation of circular motion in a vertical plane. Problems which
are still common in modern text-books are solved here for the
first time. Huygens gave also the necessary conditions for
Galileo's experiment with the intercepted pendulum (Fig. n).
This was an instructive experiment since it illustrated the
relation gh = v 2 /2 and drew attention to the fact that the
particular path makes no difference to the height achieved. This
is sufficiently explained by the figure.

FIG. 12

As has been mentioned, Huygens followed the old Aristo-
telian view that circular motion is a distinct form. This accorded
also with Galileo's view, for although he himself discovered
parabolic motion he did not include it with the other two funda-
mental motions. It seems as if both Galileo and Huygens
supposed the elliptical paths of the planets to be reducible to
simpler components. At any rate Huygens for a long time re-
served a special place for circular motion, considering it alone to
be absolute and not merely relative in nature. After the appear-
ance of Newton's Principle*, however, he came out against the

122 HUYGENS'S SCIENTIFIC WORK

idea of any absolute motion and opposed the idea of absolute
space or time. In 1694, he criticized Leibnitz for holding to the old
idea of absolute motion, but rejected the suggestion that he owed
his point of view to Newton's Principia. Mariotte also he con-
sidered was in error in attempting to distinguish the vitesses
propr.es of bodies. This, of course, raised the question of how we
may know when bodies are relatively at rest. Huygens answered
this in two ways : the bodies must be free to move and yet retain
their relative positions with respect to each other and to their
background; alternatively, by connecting the bodies by threads,
one can dispense with the background, for if rotational motion
exists, tensions will be set up in the threads.

The effect of the diurnal rotation on the shape of the earth
was, of course, a problem which Huygens could not ignore.
Kepler, in his Epitome Astronomic Copernicx (1609), seems
first to have proposed that the centrifugal force must be equili-
brated by an " attractive virtue " and Huygens followed this
lead in 1666. He calculated how much slower a pendulum clock
would go in a latitude of 45 when compared with its rate of
going at the poles. Supposing the earth to be perfectly spherical
for the purpose of the calculation this amounts to a comparison
of oscillation in a field of g units as against one g' = g - F cos a
where F is the centrifugal force acting on unit mass at a latitude
. Clearly we may put

where r = R cos a and R is the radius of the earth, T the period
of the diurnal rotation. Substituting the values for R and T, for
a latitude of o the centrifugal force is a maximum and works

out to Q approximately. This agrees with Huygens's

2o(^

estimate that the rate of rotation would have to be seventeen
times as fast for the centrifugal force to equal gravity at the
equator, but his estimate for the slowing down of the clock does
not agree with a modern estimate. His attempt to deduce the
form of the earth was, however, interesting since it laid down
useful principles for the theory of the equilibrium of fluids.
Huygens laid down the principle that a mass of fluid is at rest
only when its surface is at each point perpendicular to the
resultant force acting at that point. Later surveys carried out in

CENTRIFUGAL FORCE 1^3

the eighteenth century confirmed his general conclusions, but
not the extent of flattening at the poles which he had calculated.
For a time Cassini opposed the views taken by both Huygens
and Newton as to the form of the earth resulting from rotation.
A cartoon of the period is reproduced in Professor Capri's
edition 1 of Newton's Principia, in which the rival pictures are
shown (Fig. 13).

Newton Cassini

FlG. 13

The investigation by which Maupertuis later disproved
Cassini's conclusions gained for him the title of the "earth-
flattener ". This was, however, complimentary when compared
with the other epithets he received from Voltaire.

An account of the dispute which broke out between the
Newtonians and Cartesians on the shape of the earth was given
by d'Alembert in L'Encyclopddie of 1751-65 (Vol. VI).

1 The Mathematical Principles of Natural Philosophy, a reprint of the English
translation by Motte (1729) with some additional notes.

Statics.

In some early studies on the distribution of forces in threads
which support a number of masses, Huygens used a fundamental
principle which was closely related with the one he used so much
in Dynamics. This stated that, for a system of bodies connected
by threads and in equilibrium, a very small displacement (com-
patible with the connections) cannot cause the elevation of the
centre of gravity of the system. This idea, a variant of the idea
of virtual work, had been used by Torricelli and Pascal, but
Huygens made it the basis of a general method. He used it, for
example, in an independent proof of Jordanus's theorem on the
inclined plane (see below), and it was, of course, a form of this
principle which was employed so successfully in his study of the
centre of oscillation. To some extent Huygens followed the lead
of Galileo's Discourses but Stevin's Beghinselen der Wieghconst
(1586) and Hypomncmata Mathematica (1608) were also im-
portant and supplied him with some of his early ideas. It is well
known that Stevin regarded the impossibility of perpetual
motion to be a principle on which a treatment of equilibrium
under gravity could be based. As will be seen, Huygens's work
carried this idea further and in effect introduced the idea of the
conservation of energy in mechanical systems (see p. 154).

In the Discourses on Two New Sciences (Second Day),
Galileo had given a "proof" of Archimedes' theorem on the
simple lever and had considered also the conditions under which
a uniform beam is fractured by bending. Both of these problems
were considered by Huygens in some early studies dating from
about 1662. The defect of Archimedes' work on the lever was
that in effect he had to assume the very law he set out to
" prove ". Stevin, and later Galileo, shortened the " proof " but
neither, in Mach's opinion 1 , escaped the original difficulty; both
employed " the doctrine of the centre of gravity in its most
general form, which is itself nothing else than the doctrine of the
lever in its most general form ". Huygens did the same. His
exercise on the subject is of interest since he employed a device

1 . Mach. The Science of Mechanics (trans McCormack, 1907).

STATICS 125

which was virtually the taking of moments about an axis which
was chosen arbitrarily. Wallis, alone with Huygens, shares the
honour of introducing the idea of a moment into mechanics. In a
letter to Huygens dated January i, 1659, Wallis spoke of the
momenta of the elements of a surface about a certain axis as
forming a series in which each term was a weight multiplied by a
distance. He used the idea in finding the centre of gravity of
solids of various forms. As has been mentioned, Huygens met
Wallis in London in 1661 (see p. 45), and it seems very probable
that he was indebted to him for the germ of some of his ideas.
Much the same sort of problem as the determination of centres
of gravity was met by Huygens in evaluating the quantity Hmr
for the compound pendulum. Such important ideas were, then, in
use long before they became pan of the published work on
Mechanics. Varignon's Projet d'une Nouvelle Mecanique (1687)
was actually the first book to contain the modern idea of
moments.

In some manuscript notes on the fractures of beams Huygens
showed an approach to the modern idea of work done. Under
gravity the centre of gravity of a system will descend as far as
possible, that is, the work done will be a maximum. Huygens
used the term descensus gfauitdtis in this connection. When the
system is in equilibrium the first principle, stating that the
centre of gravity does not rise, is to be applied.

The problem! of demonstrating the equilibrium of two
smooth weights on inclined planes and connected by a thread for
the conditions

m l __ AB

^T~ """ Tar
m-j *- yo

was solved by the writer of
the work Jordani Opusculum A
dePonderositate-z thirteenth FlG - "4

century work and it was given an elegant solution by Stevin,
who simply applied the principle that perpetual motion is im-
possible. The independent solution given by Huygens is therefore
not referred to in modern works. Nevertheless, it shows clearly
the usefulness of his fundamental axiom that the centre of
gravity of a system of weights does not rise through any motion
of the weights under gravity.

126 HUYGENS'S SCIENTIFIC WORK

In the figure (Fig. 14) m l and m a are related as shown above
and are considered to undergo a small displacement. It is required
to show that the centre of gravity G is not disturbed. The point
G in the horizontal line DE must of course be such that

BD

BA

Wj

BE ~

EC

"m",

BD

BE ~

BD'

BL ~

DD'
EL

Taking DD' = EE' and D'L parallel with DE we have

EG
GD

t UZ-* LJLS LJLJ' EE

an BE = BL = EL = ~EL

Hence

EE' E'G

EL ~ D'G

so that G is also the centre of gravity of m l and m, in their new
positions.

In the history of statics, Wallis, by publishing his Mechanica
sive de Motu in 1669, with successive parts in 1670 and 1671, con-
tributed a systematic treatment which must be ranked above
Huygens's work in importance. Huygens, as has been noted,
was more interested in dynamical problems. The work of these
two men shows, however, some interesting resemblances. Both
employed the principle of virtual displacements and extended
the idea of force to all kinds of forces without distinction. Up
to this time gravitational force alone had been considered.

The Treatise on the Pendulum Clock:

The Horologium Oscillatorium. Part One

Construction and Use of the Pendulum Clock

The great Danish astronomer, Tycho Brahe, used clocks in
astronomical observations about 1580, but he was perhaps not
the first to introduce time measurement into observations in
Astronomy. Certain early Arabian astronomers, according to
Robison, used the simple pendulum for measuring short inter-
vals, but the evidence for his supposition is not very clear. No
good means really existed until the pendulum clock was in-
vented, and by the middle of the seventeenth century the need
of time measurement had become urgent. Galileo used a water
clock or clepsydra in his experiments on natural acceleration.
He clearly knew of the property of the simple pendulum : that its
period is (very nearly) independent of the amplitude of swing.
Indeed, it is very probable that this was known to Copernicus
(see p. 99). The story that Galileo discovered the property of
the simple pendulum from observation of a lamp swinging in
Pisa cathedral is not very well established, but the belief is un-
doubtedly kept alive by a passage in his Discourses on Two New
Sciences, where he said that he had often observed such oscilla-
tions in churches.

Tycho Brahe's clock was a crude instrument employing a
balance or verge escapement. Such a regulator has no natural
period, but is adjusted empirically by altering the moment of
inertia. Although such clocks were used in public buildings,
perhaps from the i3th century, they were ill-adapted for the
observatory. Tycho found it necessary to correct for the tempera-
mental qualities of his instruments at Uraniborg. It is accord-
ingly very interesting to find that that universal genius Leonardo
da Vinci, before the end of the fifteenth century, sketched, even
if he did not construct, an escapement employing a pendulum. By
the time of Galileo it seems probable that the idea of applying
the pendulum was present in many minds. The design attributed
to Galileo shows considerable mechanical ingenuity and owes
less to the old balance clock than did da Vinci's sketch.

1*7

128

HUYGENS'S SCIENTIFIC WORK

J. Drummond Robertson, in his excellent book, The Evolution of
Clockwork (1931), compared it with Huygens's design to the
latter's disadvantage. Galileo's mechanism, he wrote, " is far
more subtle and ingenious than Huygens's method of control-
ling the regular action of the old faulty escapement by the
attachment of a pendulum in place of the balance ". Neverthe-
less, Huygens's theoretical study of the problem was far more
profound and indeed left little to be added, and his escapement
also seems in practice to be much superior to that of Galileo.
His practical contributions were superseded by the invention of
the anchor escapement and later by the dead beat escapement.
The latter gives an impulse to the pendulum near its zero
position and exerts little frictional drag over the rest of the
swing. The ideal of the free-swinging pendulum was not
attained until the end of the nineteenth century.

Huygens at first connected the pendulum rod DF (Fig. 15)
to a circular balance by means of a two pronged fork or crutch.
The part DE was flexible. This arrangement was, however,
soon abandoned and the clock described in the Horologium of
1658 showed some improvement, as a comparison of the figures
(Fig. 15 and 16) will make clear. The circular balance has gone

o

Fie. 15

Fie. 16

o

THE TREATISE ON THE PENDULUM CLOCK 129

and the pendulum operates the verge with pallets V by means of
the fork Q and the wheel P engaging the pinion O. The crown
wheel is vertical. A vertical crown wheel was, however, not in-
variably used. Samuel Coster, clockmaker to Huygens at the
Hague, made some clocks with a horizontal crown wheel and one
of these, described by J. Drummond Robertson (loc. cit.), was
driven by a spring and not by weights. And although the clock
bears the date 1657, it bears metal plates or " cheeks designed to
correct the period for large swings of the pendulum. These
plates may, of course, have been added subsequently. As
J. Drummond Robertson points out, it must be supposed that
Huygens began with a horizontal crown wheel but changed this
to the vertical position in his Horologium so as to interpose the
wheels O and P in the figure. As for the use of curved plates on
either side of the pendulum it is clear from Huygens's corre-
spondence that he was uncertain whether to design die clock so as
to employ a pendulum oscillating over a small arc or to employ
the curved "cheeks". In the Horologium the cheeks are
omitted. Huygens explained to Petit that he could not,
therefore, dispense with the wheels O and P as the latter
suggested since these had the effect of restricting the arc of the
pendulum oscillations.

There is an interesting letter from Huygens to Boulliau
written in December 1657, in which he enquired about clocks
which the Grand Duke Fernando de Medici was reported to
have under construction " in order that I may know if they also
use a pendulum. A year ago yesterday exactly, I made the first
model of this kind of clock and in the month of June [1657] I
began to show the construction of it to all who asked me for it
and among whom perhaps was someone who sent news of it to
Italy ... In a very few days we shall see a much larger form of
the clock in a belfry of the village which is near the sea half a
league from here [the village of Scheveningue] . The pendulum
will be a i feet long and will weigh about 40 or 50 pounds . . . /'
The Medici Palace clock was for some time put forward as being
of earlier date than Huygens's clocks. Modern writers do not
accept it as antedating Huygens's invention, and it may in fact
have been a copy. Against this is the fact that the pendulum of
the Medici clock was rigidly attached to the escapement a
feature of both da Vinci's and Galileo's design. This was a
defect, since if the pendulum was light the oscillations could
I

130

HUYGENS'S SCIENTIFIC WORK

easily become forced and the mechanism would then resemble
an inferior balance c|pck. "... the great merits of his inven-
tion " wrote J. Drummond Robertson of Huygens's clock/' were
the free suspension of the pendulum by means of a cord or a
steel spring, with the crutch as the means of communication
between the pendulum rod and the verge; contrivances which
have continued in use ever since in all clocks with the anchor
escapement/'

FIG. 1 8

FIG. 19

THE TREATISE ON THE PENDULUM CLOCK 131

When one turns to the Horologium Oscillatorium of 1673,
one finds that Huygens gave up the two toothed wheels inter-
posed between the pendulum and the verge with pallets and
reverted to metal cheeks to render the oscillations truly
isochronous for all arcs. In 1659 he discovered the theoretical
form for these cheeks and showed that it was a cycloid arc. It
became necessary to revert to the first design employed by Coster
in 1657, and to go back to the horizontal crown wheel as shown
in the figure. Figure 17, reproduced from the Horologium
Oscillatorium, shows the flexible suspension between the cheeks
more clearly. The second figure (Fig. 1 8) shows a pendulum clock
with a tapered cylindrical pendulum and the method of using a
" maintaining weight " as invented by Huygens. This latter
device made it possible to rewind the clock without stopping it.
The cord applying the weight passed over a spiked pulley D in
the large figure. The following quotations are taken from the
Horologium Oscillatorium.

The opposite figure [Fig. 19] represents the clock seen from
the side, snowing first two plates AA, BB, half a foot long or a
little more and twenty-two inches wide, whose corners are joined by
four little columns so that the plates are distant from one another
by one-and-a-half inches. In these plates are placed the axes of
the principal wheels on both sides. The first and lowest wheel is
that which is marked C, incised with 80 teeth, and to the axis of
which is fixed the pulley D spiked with iron points to hold the
cord with the weights attached, the reason for which is given
later. The wheel C thus turns by the force of a weight; this moves
the nearest pinion E which has eight teeth and at the same time the
wheel F which has 48 teeth and is attached to the same axis. The
latter moves another pinion G and a wheel H on the same axis,
the numbers of whose teeth are the same as those of the
preceding pinion and wheel. But this wheel is of the kind that our
artisans call a crown wheel. By its teeth are turned the pinion /
and at the same time the wheel K which is on the same perpen-
dicular axis. This drum has 24 teeth and the wheel 15 which
are made like those of a saw. Above the centre of the wheel K is
placed horizontally the rod with pallets LM whose ends are sup-
ported on each side by the plates N and P separately attached to
the plate BB. In the plate NP the part Q should be noticed, pro-
jecting towards the base, through an oblong opening in which
the axis LM passes and which besides keeps in the vertical position
the axis which we have said is common to the wheel K and the
pinion /. In the plate BB a large opening is made through which
can pass the end of the rod with pallets LM, which, inserted by
its pointed end into the plate P moves thus more freely than if it

13* HUYGENS'S SCIENTIFIC WORK

were supported by the plate BB itself and were at the same
time prolonged through it; an extension is necessary so that the
crank S may be attached to it so as to oscillate with it. For this
is an oscillatory or reciprocating movement since the teeth of the
wheel K make contact in turns with the pallets LL in the
customary way and which needs no further explanation.

As for the crank S, the lower pan of which is bent back and
pierced with an oblong hole, this engages the iron rod of the
pendulum to which the bob X is attached. This rod is suspended
from above by a double thread between twin plates of which only
one, T, is visible here. For this reason we have shown at the side
a second figure intended to make dear the form of each and the
general manner in which the pendulum is suspended. It will be
necessary, however, to return to this subject and consider the true
curve of the plates.

We will now turn to the movement of the clock for we shall
explain the other parts of the figure later. It is easily seen that the
pendulum VX, when once it has been set in motion by hand,
maintains its motion through the force of the wheels driven by a
weight; and at the same time the fixed period of the pendulum
prescribes for all the wheels and consequently the whole clock,
the law and pattern of the movement. Indeed the crank, however
slight may be the movement communicated to it by the wheels,
not only follows the pendulum which moves it but also contri-
butes to that movement a short impulse at each come and go. It
maintains then a movement which without this assistance would
decrease little by little, in part at least owing to the resistance of
the air, and come to rest. Indeed, the pendulum having the
property of always following the same course, and not being
diverted from it unless its length change, the wheel K is not per-
mittedonce we have obtained the equality of which we spoke
above by means of the curved plates between which the pendulum
is suspended to go now more quickly and now more slowly
although in common clocks it often endeavours to do so; here its
teeth must necessarily pass one after another in equal times. It is
dear that the revolutions of the preceding wheels, like those of the
hands which come last, are also rendered uniform seeing that all
the parts move proportionally. Consequently if there is any fault
in the construction or, on account of a change in the timing, the
axes of the wheels turn with more difficulty, so long as this
difficulty is not enough to cause the complete stopping of the
dock there will be no reason to fear any inequality or retardation
of the movement; the clock will always measure the time correctly
or not measure it at all.

After completing his description of the mechanism, Huygens
gave directions for adjusting the clock so that it completes
twenty-four hours in the mean solar day. This is the ordinary

Equator

THE TREATISE ON THE PENDULUM CLOCK 133

adjustment of a clock for everyday use. For use in the obser-
vatory the sidereal day and not the mean solar day is the unit
required. There is about four minutes difference between these
days but the correction is not constant since the solar day is not
constant. Huygens gave a table showing the individual correc-
tions for comparing a given (solar) day with the sidereal day.

In astronomy the measurement of the time of passage of a
star across the meridian can replace the measurement of meri-
dian altitude which is more difficult and, in the seventeenth cen-
tury, the latter was always rendered somewhat uncertain through
the absence of reliable corrections for atmospheric refraction.
The history of the idea goes
back some way. To explain it,
it is necessary to refer to the
diagram (Fig. 20) which shows
the planes of the equator and
the ecliptic intersecting the
celestial sphere. As is well
known, the position of a star
in the sky is not usually re-
corded by its altitude and
azimuth because these, unlike
its right ascension and declin-
ation~or its celestial longitude
and latitude vary on the Earth's surface. In the figure O is the
position of an observer and the celestial equator and the ecliptic
intersect in the equinoctial points y and - r ^. The plane of the
ecliptic is of course the apparent path of the sun among the
stars and the equator is inclined to this at the supplement of the
angle of inclination of the Earth's axis to the plane of its orbit,
viz., 32 27'. If 5 is a star we may record its position by stating
its right ascension (arc ^M) and declination (arc MS) or its
longitude (arc ^Q) and latitude (arc QS). All these arcs are
measured in angular units except the right ascension, which is
more often measured in units of time. For since the celestial
sphere appears to revolve once in twenty-four hours, any given
star will move 15 in one hour. The interval between the transit
of the vernal equinox (the First Point of Aries) and that of a
star across the meridian therefore gives its Right Ascension in
hours. For this reason the great circles passing through the poles
P and P A and drawn at 15 intervals from the equinoxes are called

HUYGENS'S SCIENTIFIC WORK

Hour Circles. Right ascensions may thus be stated either in
hours, minutes and seconds, or in degrees.

It is obvious that when recording the observed position of a
star by its altitude and azimuth preliminary to reducing these
to one or other of the spherical co-ordinates, the exact time at
which the observation was made must be noted. This was at first
done by recording the altitude of the sun or some other standard
star. Purbach and Regiomontanus used this method in the
fifteenth century. Bernhard Walther, a pupil of Regiomontanus,
is sometimes said to have been the first to use a clock driven by
a weight for scientific purposes. His clock was useless for inter-
vals other than about an hour, however, and the first consistent
use of time intervals was due to Tycho Brahe. Tycho used quad-
rants to observe altitudes and usually found the distance along
the equator from the meridian with an armillary sphere. His use
of a meridian quadrant to observe transits required the measure-
ment of the time interval between the transit of the star and that
of the equinoctial line.

In 1667 Huygens described a method which is similar to the
well-known method of " equal altitudes " for finding the time
of a meridian passage of a star. Roemer in 1690 really established
the method of obtaining right ascensions and declinations by the
use of transit telescopes. He also set up an instrument having
altitude and azimuth circles for the observation of stars at equal
altitudes on both sides of the meridian. The clock he used bears
somfe resemblance to that of Huygens. One may take it for cer-
tain that the close association of both men with the Paris observ-
atory led Roemer to use a clock of Huygens's design. Until the
invention of the anchor escapement (1680) Huygens's clocks were
far the most accurate available and in Paris they probably con-
tinued to be preferred for many years.

While discussing this subject of time measurement in astro-
nomy one can see how the clock promised (and later supplied) a
solution to the problem of determining longitude at sea. On
March 2 1 a sidereal clock on the meridian of Greenwich agrees
with a mean time clock. For a place not on this meridian the
sidereal time must be corrected by the addition or subtraction of
9.8565 seconds for every hour of longitude difference according
as the place is west or east of Greenwich. Huygens's own direc-
tions for the use of the marine clock at sea ignored the slight
difference between sidereal and mean solar time. These direc-

THE TREATISE ON THE PENDULUM CLOCK

'35

tions were that the clock should be set going by mean solar time
at the starting point of a voyage and that the solar time at
sea (from the altitude of the sun) should be compared with the
standard time kept by the clock. To the latter it was of course
necessary to add the correction for the solar day required to take
into account the inequality of the solar days. If, then, the mean
solar time given by the clock after correction is earlier than the
observed time, the ship has moved east, if it is later the ship has
moved west. Each hour difference is equivalent to 15 of longi-
tude. Figs. 2 1 and 2-2 show certain features of one of Huygens's
later marine clocks.

FIG. 21

FIG. 22

VI

The Horologium Oscillatonum. Part Two
Oscillation in a Cycloidal Arc

This part of the Horologium Oscillatorium contained the first
thorough treatment of oscillatory motion ever given. The study
of accelerated motion in curved lines was begun by Galileo, and
it is worth while looking at his work, so neatly was it completed
by Huygens. Both of these writers treated their problems as a
branch of " geometry ". It was only after the principles of New-
ton's Principia had been absorbed that mathematicians
developed mechanics on the concepts of mass, force and impulse.
The difference is perhaps more apparent than real, for Galileo
implicitly and Huygens explicitly recognized the quantity
termed mass as distinct from weight and concentrated on velocity
and acceleration, both capable of geometrical representation.
Huygens really developed a treatment of mechanics on the basis
of work done what we now term the energy equation but this
was never fully appreciated.

In the Discourses on Two New Sciences (Third Day), Galileo
gave his classic propositions on naturally accelerated motion. He
began with the assumption that the law of acceleration is simple
and derived the well-known equations on fall from rest under
gravity. He showed, among other things, that the times of
descent of a smooth body over inclined planes of the same vertical
height are proportional to the lengths of the planes : also that
chords of the same circle are the paths over which descent takes
the same time. The latter theorem was employed by Huygens
and Galileo's proof of it may well be given here in symbolic
form.

In Figure 23, B and C are any two points on the circumference
of a circle, centre O and radius r. AF is the vertical diameter,
AD and AE the vertical heights of AB and AC, and AI is
the mean proportional of AD and AE. Putting AD~x lt
AE SB # 3 , AI = ra> we have, therefore, m = >/*i* 3 .

136

OSCILLATION IN A CYCLOIDAL ARC

Now AC' = x* + CE 1 A

and CE? = CE.EC' = r 3 - OE* <

Hence

'37

= AT a 2 + r* -

/4C'
/4B a

n x~ m

But = .

m

FIG. 23

For the times of descent along the inclined linfcs we have the
equations _ 2

where ^ and a 2 are the accelerations produced.
But

sn

AB

Hence

AC. sin Q!
t* AB. sin a 2

^

i.e.

" AB 2
AC

~AB

m

and by (i) above = i.

Galileo actually proved the relation (ii) in a separate proposi-
tion.

Galileo pointed out that the speed acquired in descent over
an inclined plane is always such as would enable a body to re-
ascend another plane of equal vertical height. His experiment
with the intercepted pendulum
illustrated this point (see p. 121).
He went astray, however, in sup-
posing that a circular arc is the
path of quickest descent under
gravity.

It is usually supposed that
Galileo considered the simple
pendulum made isochronous
oscillations in all arcs and that

138 HUYGENS'S SCIENTIFIC WORK

the period was quite independent of the amplitude. This is cer-
tainly assumed in the Discourses, where Galileo comments on the
fact that descent from A to B over the inclined plane (Fig. 24) takes
longer than descent along a circular arc CB. He then added:
" As to the times of vibration of bodies suspended by threads
of different lengths, they bear to each other the same proportion
as the square roots of the lengths of the threads . . . ". There
is no record in his writings to show that he observed the dis-
crepancy of the periods of large and small swings. The Floren-
tine Academicians, however (Essayes of Natural Experiments
made in the Accademia del Cimento, translated by R. Waller, 1684),
said that Galileo observed the " very near equality " of the
swings. Mersenne, in his work Les Nouvelles Pensees de Galileo
touchant Les Mecaniques et la Physique, said that Galileo did
not observe the discrepancy. " If the author had been more exact
in his trials," he wrote, he would have noticed it. His regrets ap-
pear somewhat out of place, however, when one considers the
fertile ideas Galileo drew from his study of the pendulum; an
exact relation between the amplitude and the period could have
been only a hindrance at this stage. Enough is known of Gali-
leo's methods for us to be sure that an element of idealization
came into his treatment of the pendulum, so that he perhaps
intentionally ignored small experimental deviations from the for-
mulated law. The laws of nature, he believed, must be charac-
terized by their simplicity.

Huygens did not study the simple pendulum exhaustively.
After showing that the oscillations could be regarded as isochro-
nous only when the arc was small he turned his attention to the
problem of the tautochrone, that is, the curve over which all
oscillations take the same time under gravity. He showed that
the cycloid satisfies these requirements, being the curve traced by
a point on the rim of a wheel which rolls along a horizontal sur-
face. It was left to Lagrange and Laplace to complete the study
of the simple pendulum and derive some form of relation be-
tween the period and the amplitude. The former's Mtcanique
Analytique (Chap. 2) (1788) and the latter's Mecanique Celeste
(Chap. 2) ( 1 799) should be consulted.

Throughout this part of the Horologium Oscillatorium Huy-
gens's debt to Galileo is very clear. The first nine propositions,
in fact, are really a clear resume of Galileo's work on natural
acceleration with some small additions. A good deal of ground

OSCILLATION IN A CYCLOIDAL ARC

139

had to be cleared before Huygens could get to grips with the
central problem. Some geometrical properties of the cycloid, a
method of drawing a tangent to the curve, and the setting of
limits to the length of an arc of a circle occupy the next few

FIG. 25

pages. Then in Proposition 22 Huygens showed quite simply
that if two cycloidal arcs of equal vertical height are considered,
descent from rest is quicker over the steeper curve. In Fig. 25,
BD and EF are the arcs, equal in vertical height h but one less
steep than the other.

By a property of the cycloid the tangent at any point L is
parallel with the chord NA of the generating circle. L and M
are taken such that L is the same vertical height below B as M is
below E. Transferring the arc EF to the position ef the point M
is at m, level with L. Since the inclination of the tangent at ra is
given by the chord OA and for all points such as L and m the
inclinations of the tangents to the steeper curve are themselves
steeper, the time of descent must be shorter over BD than over ef.

In the next proposition Huygens compared the time of

7?
F

FIG, 26

140 HUYGENS'S SCIENTIFIC WORK

descent over a short segment of a tangent to the cycloid with that
over an inclined plane of equal height. Supposing a body to be
released at B in Fig. 26, and to descend along the arc BG, he com-
pared the time of descent over MN with the speed attained at G
with the time of descent over OP with the mean velocity acquired
during descent over the distance BI, the line BI being the tan-
gent to the cycloid at B. Using Galileo's relation v 2 = igh for
vertical descent, Huygens took the velocity of a body after des-
cent over BI to be \/2g. ir or 2 fgr and the velocity at G to be
One half of the former is thus

TK

111118

By similar triangles
FA FH

FH ~ FE
FA _ FH*

and ~ FF

FA FH

_ _

~FH ~ FE ~

l^ FX

Hence = jg-

t t MN ^ Vl FX MN

Now _ = ^ . ^_ = __ . _.

The remainder of the proof consists of showing by geometry
that

MN FH

Hence

OP " HE '
t, FX HX ST

t t ~ HE ~ HE ~~ QR
The last equality follows since HE = HX cos 6

QR = STcos*.

It appears a somewhat curious relation to set out for but the
time of descent over a cycloid arc is later to be obtained by
reference to a relation between the time of descent over a series of
tangents ST, and the time of fall along an intercept on the axis
FA. Unfortunately, the following theorem is unsuitable for re-
production here in its original form. Anyone who looks up the
work in the original will see that here is an interesting method
of exhaustion pushed to its limits.

OSCILLATION IN A CYCLOIDAL ARC 141

In Fig. 27 a large number of tangents are drawn to the cycloid
and to the circle. It is required to show that the time of descent
t l9 along an arc BE, is to the time of descent f a , along the tangent
BI (with a uniform speed equal to the mean speed acquired over
Be) as the arc QH is to QG. The intercepts made in BI being sup-
posed equal, the time interval S/ 2 for motion along each is a fixed
quantity. The relation with the previous theorem is then best
shown by taking t l to be the sum of the intervals 8*1, required for
descent over the tangential elements M^N^ M 2 N 3t etc.

FIG. 27
The equation of the previous theorem is then written in the

81, = S lTl

dt* QR

where QR is the interval between the parallels measured on the
diameter QA. Dividing this equation by the number of elemen-
tary tangents, n, we get

and there are n of these relations altogether. Adding these
equations we get

_

n^ a = QG~'

In the limit, when the elementary tangents are very small and
very numerous, 257 = QH, 23^ = ^ and we have

!i . OIL

* "" QG-

Huygens was then able to complete his demonstration by

142

HUYGENS's SCIENTIFIC WORK

showing that the time of descent from any point on a cycloid
to the lowest point has, to the time of descent along the axis,
the ratio of the semicircumference of a circle to its diameter.

FIG. 28

Applying the previous theorem to the case of descent to the
lowest point

^ = ar ^^
t M

where t M represents the time of descent with the mean speed
acquired over the tangent BG. This, however, is the same as the
time for naturally accelerated fall along BG. Now BG is equal
and parallel with EA (by a property of the cycloid) (Fig. 28).
Hence :

*BA _ mrcQHA _ ^r_ __ ic

1 ~ Q/T ~

IT

By Galileo's theorem (p. 137), / EA may be replaced by J DA since
EA and DA are chords of the same circle. This establishes the
tautochronism of the cycloidal pendulum.

Huygens did not include in the Horologium Oscillatorium his
proof that in a cycloidal pendulum the restoring force is propor-
tional to the arc of displacement. This short addendum is,
however, important, since Huygens was the first to give the
mathematical theory of simple harmonic motion. Leibnitz wrote
to Huygens in March 1691 : " M. Newton has not treated the laws
of the spring; I seem to remember having heard you say on an-
other occasion that you have examined them, and that you had
demonstrated the isochonism of the vibrations." We may sec

OSCILLATION IN A CYCLOIDAL ARC

143

for ourselves that the fact that the cycloidal motion is simple har-
monic follows from Huygens's previous discoveries. He seems
to have arrived at the conclusion about 1673.

FIG. 29

In Figure 09 the cycloid arc AC is divided at some point B.
" Then," Huygens wrote, " the component of gravity along the
plane tangential to the curve at A and the component at B is in
each case proportional to the slope of the tangent ", i.e., in
modern terms :

Force down plane at A
Force down plane at B

gstna

g sin
EM

\
PC

EC
OQ

PN
PC

OC
PC
OC

OQ

But, by a lemma to the theorem on page 139, Huygens showed by
simple geometry that

PC _ EC

OC ~ PC'

Hence : Force down plane at A __ EC
Force down plane at B PC

It was not difficult to show that EC is to PC as the lengths of the
respective cycloidal arcs AC and BC. This relation may readily
be demonstrated by the use of modern methods but Huygens's

144 HUYGENS'S SCIENTIFIC WORK

proof was not presented in a formal manner. With it the theorem
on simple harmonic motion is completed : the acceleration at
any point on a cycloidal arc is proportional to the length of the
arc measured from the lowest point. It is thus clear from his note-
book that Huygens cleared up most of the obvious problems
which are raised by a study of oscillatory motion. Such prob-
lems can only be dealt with satisfactorily by means of mathe-
matics.

It must be admitted that Hooke, in England, earlier than
Huygens, had commented on the conditions required for simple
harmonic motion. Birch's History of the Royal Society (i 756-57)
contains a passage by Hooke dated Nov. * i , 1666, on this subject :
" The equality of duration of vibrations of differing arches or
lengths depends upon the figure of the curve-line, in which the
body is moved; which figure being for a very great part near
the same with that of a circle, it follows that, the motion in
differing arches of the same circle will be very near of equal
duration/' Hooke's attempt to show mathematically that this
followed from (Galileo's) mechanics was rather feeble, however.
Even Brouncker, president of the Society and a much more able
mathematician, made little headway with this subject. Yet both
Brouncker and Hooke started where Huygens started and with
the same fundamental conceptions. Hooke's Potentia Restitu-
tiva, published in 1678, contained a further reference to the sub-
ject and this work is usually regarded as being the starting point
in the history of simple harmonic motion. It contains the law
" ut tensio sic vis " and many deductions drawn therefrom.
Nevertheless his contribution cannot be compared with that of
Huygens, whose work was carried out quite independently. The
properties of springs were well understood by Huygens in 1675,
for this is the date of his invention of the spiral-spring regulator
for watches. A further illustration of his grasp of the essential
condition may be found in his experiments on the use of a tri-
filar pendulum. This consisted of a heavy flat ring suspended
by three equal vertical cords from three equidistant points. The
oscillations of this pendulum were of a torsional character and
he hoped that it would prove superior to the spiral spring, especi-
ally in regard to the effects of temperature. It did not appear
to matter how the restoring force was set up whether through
gravity, elasticity or magnetic attraction he remarked.

VII

The Horologium Oscillatorium. Part Three
Evolutes and the Measurement of Curves

The occasion may be taken for a brief mention of Huygens's
mathematical work. This seems to be divisible into two parts.
In one part of his mathematical work Huygens may be said to
have belonged to the classical schools of Archimedes, Hippo-
crates of Chios and Eudoxus rather than to the period of Des-
cartes, Newton and Leibnitz. He was, for example, greatly inter-
ested in the three great problems of antiquity, those of squaring
the circle, trisecting an angle and duplicating a cube, and he
was in the classical sense the outstanding geometer of his age.
The impress of classical antiquity lies on most of his early work.
His earliest published work, Theoremata de quadrature hyper-
boles ellipsis et circuli, ex data portionem gravitatis centra (1651),
was undoubtedly inspired by Archimedes' work in hydrostatics
dealing with the flotation of certain geometrical figures. It was
in a sense a continuation of Archimedes' De /Equiponderantibus,
and in finding the area of a given segment of a hyperbola,
ellipse or circle, less than the whole figure, Huygens used the
classical method of exhaustions. In the form of the " method
of indivisibles " this method had been extended by Kepler and
Cavalieri, the latter's work being especially influential. Huygens
took the opportunity of showing up the fallacy in some work
on quadratures by Gregory de St. Vincent, a contemporary Jesuit
mathematician who appears not to have understood the newer
developments.

Huygens began his treatise by determining limits to the area
of a given segment of each of the above figures. He then located
the centres of gravity and worked out some theorems, in particu-
lar a theorem concerning a relation between the length of an
arc of a circle, its chord, the radius and the distance from the
centre to the centre of gravity of the segment. This paved the
way for an examination or " Excursus " criticizing de St. Vin-
cent's work. This excursus occasioned a dispute which lasted for
ten years and in which several mathematicians were involved.
It has not left any important results, however.
* H5

146 HUYGENS'S SCIENTIFIC WORK

In January 1 652 Huygens began to study problems which lead
to equations of the third or fourth degree. These " solid " prob-
lems also arose, very probably, from a study of Archimedes'
writings. In the course of his work Huygens (1654) carried out a
new determination of the circumference of a circle. In the form
of a determination of *, van Ceulen, early in the century, had
made a new contribution to this ancient problem. Snell, in his
Cyclametricus of 1621, narrowed the limits set by Archimedes in
his original investigation but made use of propositions which
were not rigorously proved. In all this, however, there was little
if anything which could be called new in Huygens's work. The
more interesting parts of his studies on the problems bequeathed
by Archimedes are concerned with physics rather than with
mathematics. In continuing the latter's studies of flotation, for
example, he was able to make use of his fundamental principle
that the centre of gravity of a system takes up the lowest posi-
tion consistent with the restraints. In the first four theorems of
his De Us quae liquido, supernatant (1650) he deduced in this
way the horizontal surface of a stationary liquid, the equilibrium
of floating bodies when the density is equal to that of the sup-
porting fluid, and the celebrated law of Archimedes for the case
in which the density of the solid is less than that of the liquid.
In the other part of his work to which we may now turn
Huygens was interested less in continuing the study of classical
problems than in new and original developments. Reference has
been made on more than one occasion to Huygens's fruitful in-
terest in the cycloid. This curve may be taken as a starting point
in reviewing briefly his important work on evolutes. Descartes
influenced Huygens very much in this connection, for he had
always insisted that any curve whose mode of generation could
be clearly conceived belonged to Geometry. Descartes accord-
ingly took all curves formed by the intersection of two moving
lines, the rates of movement of which had a known ratio.

Huygens did not commence his own researches on the
cycloid until twelve years after Mersenne, in 1646, gave him his
first information about publications on this interesting curve.
Pascal's "Dettonville" problems, as has been mentioned, aroused
Huygens for the first time. The problems were to find the area
of a half-segment EBF of a cycloid (Fig. 30), the position of its
centre of gravity and the volumes of the solids produced by
revolution of the segment about BF and about EF.

EVOLUTES AND MEASUREMENT OF CURVES 147

Huygens at first found the area EBF and thence the area of
the whole segment EBO and he found the distance of the centre
of gravity of the segment from the base EO and deduced the
volume of the solid of revolution about this base. Pascal then
suggested the calculation of the centre of gravity of the half-
solid of revolution of ABD about AD. Huygens obtained a par-

A D

FIG. 30

tial solution to this but found the work so difficult that he
doubted if all Pascal's problems could in fact be solved. In pass-
ing it may be noticed that Huygens improved on the ordinary
proof required for the method of drawing a tangent to a cycloid.
In the preceding figure a tangent to the cycloid at E must be
parallel with the chord BG of the generating circle an interest-
ing property which was given in Schooten's edition of Descartes's
Geometria.

Pascal's proofs for the Dettonville problems compare favour-
ably with the work carried out by Huygens in being altogether
more elegant and at the same time more general. Wallis, how-
ever, complained about Pascal's methods in his De Cycloide of
1660, in which solutions to the Dettonville problems were
obtained by means of Wallis's " arithmetica infinitorium ". A
dispute between Wallis and Carcavy ensued, Huygens acting as
intermediary, but it does not seem to have been of great signi-
ficance. The really interesting feature of the whole episode is
that it started Huygens on his study of evolutes.

The idea of the evolute of a curve may be explained by refer-
ence to the parabola. This curve (Fig. 31) may be described as
the locus of points equidistant from a given point F (the focus)
and a given line XY (the directrix). With the exception of the
circle, all curves show varying curvature; whereas all normals to
the circle intersect at the centre, the normals to other curves
intersect at a series of points which generate another curved line.

148

HUYGENS'S SCIENTIFIC WORK

FIG. 31

This is termed the evolute. This
will be clear from the figure in
the case of the parabola.

The evolute may in fact be
defined as the envelope of the
normals to the given curve.
Huygens no doubt saw, in his ex-
periments with a simple pen-
dulum, that the bob could be
made to describe a variety of
different arcs according to the
shape of the curved " cheeks "
between which it was suspended.

Propositions 5 and 6 of the third part of the Horologium Oscilla-
torium contain the discovery that the evolutes of a cycloid are
themselves cycloid arcs. The proof is of course geometrical. In
Fig. y. the arc AF is equal in length to the arc AC. By establishing
this Huygens was able to rectify the curve and show that a cycloid
is four times the length of the diameter of the generating circle.
It is not considered necessary to recapitulate his method here.

Huygens is remembered as the discoverer of the evolutes of
a cycloid, but it must be pointed out that he dealt also with the
evolutes of a parabola (proposition 8), of an ellipse and of a
hyperbola (proposition 10). He also showed how to rectify
curves for which the evolutes are known. This work has not been
given much attention, possibly because of the more important
general methods of quadrature worked out by Wallis. Huygens

EVOLUTES AND MEASUREMENT OF CURVES 149

himself does not appear to have seen that the theory of evolutes
would find its most useful application in a field in which he him-
self was the first to explore: the theory of focal lines in geometri-
cal optics.

To the cubical parabola and cycloid Huygens later added the
curve known as the cissoid to be included among curves whose
rectification could be accomplished. The French physician,
Claude Perrault, set the problem: to determine the path in a
fixed plane of a heavy particle attached to one end of a taut string
whose other end moves along a straight line in that plane.
Huygens and Leibnitz studied this problem in 1693, and worked
out the geometry of the tractrix. It was at this time that
Huygens solved the problem of the catenary and determined the
surfaces of certain solids of revolution.

After his return to Holland in 1681 it seems as if Huygens
resorted to pure mathematics and gave less place to physical work.
About this time the infinitesimal calculus acted as a great stimulus
to mathematicians, and it is not surprising, therefore, to find
that Huygens's notebooks contain a great deal of work of a more
analytical character done after his return to the Hague. A review
of all his mathematical work would be a most valuable contribu-
tion to the literature of the history of mathematics for, over a
long period of years, from his earliest work on the circle there is
a range of subjects, many of which bear a logical relationship,
including work on maxima and minima, which links up with the
work of Fermat and the English mathematicians of the period
and culminates in the work of Leibnitz and Newton.

VIII

The Horologium Oscillatorium. Part Four
The Centre of Oscillation of a Compound Pendulum

Passing reference was made on page 99 to the existence of action
and reaction in the case of connected bodies. Galileo was unable
to account for the behaviour of two small masses suspended along
one thread (Fig. 33). He may have supposed that it should be
possible for the masses to swing in unison, whereas in fact two
independent vibrations exist in such a system. This is
a difficult problem and there was no important practical
reason for pursuing it. A more urgent problem was to
calculate the period of a given compound pendulum,
that is, a rigid body suspended so as to oscillate about an
axis which passes through it. This is the type of pen-
dulum employed in the pendulum clock, and the
problem is to find /, the distance from the axis of oscilla-
tion to the centre of oscillation. The latter is the point
at which the entire mass would have to be concentrated
in order to obtain the equivalent simple pendulum, that
k is, that which possesses the same period of oscillation.
_, The scientific principles of clock construction could not

" ^ be said to be known until this central problem was solved.
What was more important was that this particular problem opens
up the whole subject of dynamical systems.

Huygens was early acquainted with the problem by Mersenne
and supposed that it emanated from him. The problem had,
however, a longer history. The work In Mechanica Aristotelis
Probletnata Exercitationis (1621), by Baldi, contained some
erroneous suggestions on the centre of percussion and the sub-
ject even seems to have been discussed by certain Greek writers.
A modern illustration of the centre of percussion would be the
point on a cricket bat at which the ball must be struck to secure
the maximum effect with least effort. The centre of percussion, if
it exists, is the same point as the centre of oscillation, a relation
which was found empirically by Mersenne and proved theoreti-
cally many years later. The interest shown by mathematicians like
Descartes and Roberval in this problem must be attributed

OSCILLATION OF COMPOUND PENDULUM 151

in some degree to the practical value of the subject in the
design of sword blades. None of these, however, achieved any
success. In a letter to Mersenne in 1646, Descartes dealt at some
length with certain special cases of the centre of oscillation
those of a long rod, a plane triangular figure and others. He
clearly thought that the problem could be reduced to one in
statics that of determining the centre of gravity of certain
solids and planes. The work was surprisingly slipshod and
amounted to little more than a series of ingenious guesses and
no proofs could, of course, be given.

Although Huygens's experiments on the centre of oscillation
may have been carried out from as early as 1646, when his corre-
spondence with Mersenne started him on the study of several
problems of mechanics, his theoretical studies can hardly be
dated earlier th^in 1659. He appears to have employed experi-
ment much as Galileo did, and his procedure was in fact an
excellent illustration of the inductive method. Starting with the
simplest case of a linear rigid pendulum, he proceeded to study
the oscillation of laminae oscillating in their own planes. For
these cases he was soon successful in finding a way of calculating
the length of the equivalent simple pendulum. Expressed in
modern form this amounted to an application of the relation :

mh

where / is the moment of inertia about the axis of oscillation,
m is the total mass and h is the distance of the centre of gravity
from the axis of oscillation. The concept of the moment of
inertia originated in Huygens's work, but it was given this term
later by Euler. Huygens, however, discovered the important
theorem which relates the moments of inertia of a lamina
about two axes perpendicular to each other. All the laminae
considered had regular shapes possessing an axis of symmetry.
From these cases he passed on to the study of solids of revolution
produced by rotating the laminae about the axes of symmetry. For
this a method of evaluating Srar 3 for the relation / = 2mr 2 had
to be devised and this part of the work makes the hardest reading
for the modern reader who is accustomed to solve such problems
by means of the integral calculus.

As might be expected, Huygens started with certain problems
left unanswered by Galileo and he was able to solve them by the

'5*

HUYGENS'S SCIENTIFIC WORK

application of his theorems on impact. As will be seen the
subject of moments of inertia came out of the solution obtained
in this preliminary work.

D F

FIG. 34

In Fig. 34, an inflexible weightless bar carries two masses, D
at the lower end, and E at some point in AD. Given the masses
E and D, and the distances AD and AE, the problem was to
find the centre of oscillation. The great difference here is that
Huygens made the connection rigid, whereas in Galileo's
problem the masses were connected by a light thread.

Huygens's treatment was to suppose the pendulum with-
drawn through an arc to the position ABC and then released.
After passing through half of a swing the masses are again at
the positions E and D and possess speeds which are different from
those which they would acquire as the bobs of two separate
pendulums of lengths AD and AE. Huygens supposed the
two masses at this point to collide with masses which are respec-
tively equal to the given masses but not connected. There is no loss
of momentum and if the masses are all perfectly elastic the bodies
F and G will acquire the speeds possessed by D and E, the latter
being brought exactly to rest by the impact. Huygens then applied
the equation connecting potential and kinetic energies to these
masses. This manner of treating the problem was quite new.
He supposed that the speeds acquired by F and G are such that
they can rise against gravity to the positions N and V. The
essential point in this process was that the centre of gravity of
the masses could not ascend beyond its original height when the
pendulum was in the position ABC.

OSCILLATION OF COMPOUND PENDULUM 153

In Fig. 34 is shown the equivalent simple pendulum HK of
length x. If we put AD = a t AE = b t the mass of D = m l9
that of E = m a and their speeds at the lowest points respectively
v l and z>2 we have

speed of D v l _ arc CD AD _ a
speed of K = V = 5rc?K * HK " x '
where v is the speed of the bob K through its lowest point. If
the height CS is put equal to d, the height QP is given by

QP = d.Z

It has to be borne in mind that the equivalent pendulum swings
with the same period as the compound pendulum. The height
RN to which the masses F can ascend is such that

QP

Now # a

Hence the height which the mass F can attain is ad/x and for
the height MV, for the other mass, we obtain in the same
manner b 2 d/ax.

The centre of gravity of the two masses will then attain a
height which can be calculated quite simply: the work done in
raising the masses through their respective elevations NR and
M V is equated to the loss of potential energy in the descent of
the masses through CS and BO. These heights may then be ex-
pressed in terms of x, the length of the equivalent simple
pendulum.

F.NJR + G.MV = D.CS + E.BO

ad b'd , bd

or m l ~ + m a = m^d + m *
x cix **

i.e m^d + m^b 2 d = m^dax + m^bdx.

If this treatment is extended to a uniform rod, considered as
composed of contiguous masses, the general formula

Smr*

154 HUYGENS'S SCIENTIFIC WORK

is obtained. The quantity Smr* was thus introduced into
mechanics. In this work Huygens made effective use of the
idea of work done against gravity, as can be seen. Unfor-
tunately, his ideas were not made explicit and he did not for-
mulate a general method. His published work, however, .showed
some improvement on this original form of the " direct " method.
The device of supposing an impact with equal masses which are
not connected was abandoned and, in the Horologium Oscil-
latorium, Huygens simply supposed that the separate masses
constituting the pendulum were freed from their connections at
some point in the swing. The mathematics remains unchanged.
It is important to appreciate exactly what Huygens had done
in this piece of work, for it has a bearing on the development of
the concept of energy in physics. The central idea lay in com-
bining his fundamental principle that the centre of gravity of
the masses cannot ascend as a result of a displacement occurring
under gravity with Galileo's relation connecting the speed
acquired in falling with the square root of the height. In the
Horologium Oscillatorium the speeds of suspended particles in
any point in the path were compared with the corresponding
heights of descent, a procedure which really amounted, in com-
bination with the conservation of vis viva, to the application of
the law of conservation of energy in mechanics. It is possible
from this starting point to obtain an equation showing the con-
stancy of the sum of the kinetic and potential energies for an
isolated system, namely :

T + V = H

in the form given by Lagrange. If any doubt existed as to the
significance of Huygens's contribution to the subject of energy
it may fairly be stated to be removed by a later statement of his
ideas. In his MS. of 1693, two years before his death, he wrote:
"In all movements of bodies whatsoever, no force is lost or
disappears without producing a subsequent effect for the produc-
tion of which the same amount of force is needed as that which
has been lost. By force I mean the power of raising a weight.
Thus, a double force is that which is capable of raising the same
weight twice as high." The word for force in this passage was vis,
a word which, like potentia, was used in the seventeenth century
in the two senses of force and energy. Unfortunately, the defini-

OSCILLATION OF COMPOUND PENDULUM 155

tion is ambiguous but it would fit the notion of work or energy
rather better than that of force in the Newtonian sense. The
passage clearly reduces to this: there is something about a
moving body which enables it to effect changes in the state of
either itself or other bodies, and these results are quantitatively
related with the cause which is the force or energy involved. The
same idea was far less clearly expressed by Leibnitz /about this
time : "... it seems necessary to admit in bodies something
other than magnitude and velocity unless we are willing to deny
to bodies all power of action ".

To return from this digression to the contents of the
Horologium Oscillatorium : Huygens found it impossible to
apply the " direct " method to determine the centre of oscillation
of suspended solids. The location of the centre of oscillation of
a suspended sphere, for example, cost him much time and the
method given in this work is long and difficult. His ideas appear
altogether strange to the modern reader and an explanatory note
will not be out of place at this point. To understand how
Huygens's ideas took shape it is of interest to know that he read
a work entitled Tractatus Physicus de Motu locali by one
Mousnier, a pupil of Pere Honori Fabri, which came out in 1646,
and in this there was a novel attempt to solve the problem
of the centre of percussion in the general case. Little seems to
be known of Mousnier, and his ideas may, to a large extent, have
been derived from Fabri. Mousnier used a concept which he
termed the " impetus " of motion. By this he meant a quantity
varying as the product my for a particle in motion and he applied
this to the elementary parts of an oscillating body as follows :
considering the oscillation of a plane surface about an axis in its
own plane, to evaluate the total " impetus " of all the elements,
Mousnier took lines perpendicular to the surface and these
represented by their lengths the speeds of the elements (to which
they were normal) at the lowest points in their paths. This con-
struction generated a wedge-shaped solid. For laminae of simple
form the volume of the wedge could be calculated. Mousnier
did not know, as Huygens did, that the centres of percussion and
oscillation are identical. The identity had not been proved at
that time and it was doubted by some, notably by Roberval, who
thought the two points were only approximately in the same
position. Proceeding by experiment, and at first inductively,
Huygens was able to find the equivalent simple pendulum

156 HUYGENS's SCIENTIFIC WORK

for a lamina oscillating in the way stated, and this opened the
way to deal with suspended solids. The problem in every case is
to evaluate the expression Smr 2 .

To take a simple case, the lamina ABC (Fig. 35) oscillates
about the axis EAE which is tangential. Huygens proceeds
to construct a wedge-shaped solid
on the lamina as base : a second
plane is taken at 45 and a
generating line DB, perpendi-
cular to ABC, moves round
the boundary of the lamina
tracing the projection on the
inclined plane. The centre
of gravity of the enclosed solid
can be found if the base is of
simple geometrical form, and a p IG -

plane of symmetry meets the

axis of oscillation perpendicularly at A. From the centre of
gravity, X, Huygens drops a perpendicular XL to the base and
gives the name subcentric to the straight line AL. This is only
a term for the distance of the centre of percussion from the axis.
The wedge is in fact a geometrical representation of the ex-
pression fv.da where da, in the ordinary calculus notation, is an
element of the area of the lamina and v is its linear velocity at
the mid-point of the oscillation. If we write dm in place of da it
becomes more clear that the wedge represents the summation of
the linear momentum of the lamina as it passes through the
mid-point of its swing. By determining the point L, Huygens
was in fact finding the position at which the total mass M would
possess the same momentum. Putting AL = /, we may, for
example, take the simple case where, for the lamina, we have
simply a uniform rod. The result for this case is already known
by the direct method to be

/ =

jydy fcy

(this is a modern form of the expression given before). The same
result may also be obtained by the new geometrical method, for
the point X would obviously lie at the point of intersection of
the medians of a right-angled isocceles triangle. It follows from
this that AL = 4B(Fig.36).

OSCILLATION OF COMPOUND PENDULUM 157

In the case of a lamina a method of finding the volume of
the wedge was required. Huygens showed that this could be done
by means of a simple relation which may be expressed :

volume = (area of baseXdistance AF)
where F is the centre of gravity of the lamina (Fig. 37). Huygens's

FIG. 36

FIG. 37

T

lO

method is in effect to treat the determination of Srar 2 about the
axis EAE as a geometrical problem. He does not explain how he
made his discoveries but shows how 2mr 2 for a lamina of simple
shape can be found for several different axes of oscillation
including an axis through the centre of gravity or at a fixed
distance from it.

When he took up the problem of
finding the moment of inertia of a sus-
pended solid his procedure showed an
essential resemblance with that employed
when such problems are solved by means
of the calculus. In this case the regular
solid is divided by planes chosen in a
suitable direction so as to form a series
of laminae of simple geometrical form.
The expression y*.da is then integrated
between the limits of y, the distance of
any element from the axis of oscillation
after expressing da in terms of the co-
ordinates x and y . The integration is then
required for an expression in terms of y
and dy. This is scarcely a satisfactory
statement about Huygens's method, FIG. 38

i

158 HUYGENS'S SCIENTIFIC WORK

especially since he lacked a general method of summation, but it
does not appear possible to describe his procedure in greater detail
and the reader who is interested is referred to the German transla-
tion in Ostwald's Klassiker der Exakten Wissenschaften, No. 192,
to which are appended extensive notes in more modern form.
Huygens's method could not very well be applied to more than
a few regular solids.

The centre of oscillation C is in all cases lower than the
centre of gravity G of the suspended solid. If / is the distance of
the former from the axis of oscillation O and r the distance of
the latter, Huygens showed how (/ r ) could be calculated in
certain cases. He was impressed by the discovery (1664) that the
product (/ r )r is constant and he later termed this the
" rectangulum distantiarum". For parallel axes of oscillation
the equation :

(/ - r )r = (V -r > ' (i).

could be applied, the second set of symbols referring to the
second position of the pendulum. This enabled him to calculate
exactly the effect of lengthening the suspension. If on the
second occasion the axis of oscillation is made to pass through
the point which was formerly the centre of oscillation it follows
that the new centre of oscillation will be at the position of the
former axis, for, from Fig. 38 :

'.' = I - r Q (ii).

and since I r _ r/

/'~-~~V ~ r

I' r.' = r

and from (ii) /' = r + I r

= /.

This means that the compound pendulum has the same period
in these two positions. The idea of the reversible pendulum was
of course applied very successfully by Kater. The Kater
pendulum is in fact the best laboratory method for the deter-
mination of the acceleration of gravity, since the distance
between the two knife edges on which its period is the same can
be accurately measured.

No theorem requisite for the complete theory of the
pendulum clock was omitted by Huygens. The initial difficul-
ties being cleared away he completed his treatise with a consider-

OSCILLATION OF COMPOUND PENDULUM 159

ation of the effect of moving a small weight or rider along the
rod of the clock pendulum. The latter consists of a rigid rod
carrying a heavy sphere at its lower end. To find the centre of
oscillation of such a pendulum it was necessary to combine the

values of - -- for the rod (of length L) and the sphere found

separately. Taking the respective masses to be m x and m a he
showed that Smr a = ^rnJJ for the rod and 2mr 1 = m J L a for the
sphere, L being measured from the centre of the sphere. The
corresponding values for 2mr were ^m^L and mJL. Hence for the
combination

A simplification is introduced into this work in that the radius of
gyration of the suspended sphere is not L but (i/jR* + L*^
where R is the radius of the sphere. Huygens proceeded to
calculate the centre of oscillation for the same pendulum carry-
ing a small spherical rider at a given position on the rod. He
showed that for a given alteration in the period there were two
positions in general for the rider.

At the end of the work, Huygens proposed a unit of length
based on the pendulum. The standard foot, or pes horarium,
was to be one-third of the length of a simple pendulum which
beat seconds at Paris. The size of the bob would, of course, be
immaterial since for a sufficiently large bob, which could not be
regarded as a simple pendulum, the centre of oscillation could
now be found. The objections to this unit were the variation of
g, the acceleration of gravity, with latitude and the inaccuracies
which must arise in measuring the length of the thread. It was,
however, an ingenious suggestion designed to overcome the
objections which apply to the dependence on a bar of standard
length.

In spite of its forbidding appearance in the original,
Huygens's work in this part of the Horologhim Oscillatorium
contains some strikingly original ideas. The idea of work or
energy, implicit in some of his work on impact, turns up again
here in a more precise form. The idea of vis viva indeed comes
from the work on impact while that of work done is involved in
the direct method of finding the centre of oscillation. Someone

l6o HUYGENS'S SCIENTIFIC WORK

indeed was bound to derive the last idea from Galileo's equations
of motion which, multiplied by m give (in modern symbols) :

mv = mgt = Pt (i).

ms = imgf = IP? (ii).

mgs = \mv* (iii).

If Huygens's work had been more promptly published, and
still more, if he had not cast all he wrote in geometrical form, the
third equation might sooner have entered into mechanics. In
the work on moments of inertia, as in the contribution to the
subject of energy, Huygens has not been duly recognized. Like
Newton's Principia, the Horologium later came to be regarded
as pretty hard going for readers who were accustomed to analy-
tical methods.

IX

The Cause of Gravity

The idea of a force which acts towards the centre of the
Earth dates from the earliest times 1 . Aristotle's doctrine that
the elements, excepting fire, have a tendency to take up their
" natural " places was a modified form of the idea and it was
reproduced by the scholastic commentators. A clearer statement
was given by Copernicus, who wrote " The Earth is spherical, for
all its parts strain towards its centre of gravity." Much later
Mersenne defined the " centre of the universe " as the point
towards which all heavy bodies tend in straight lines. Gilbert
(1600) attributed the action of gravity to other bodies besides the
Earth, but he did not regard it as universal. The reason for
motion under gravity was for him " a substantial form, special
and particular, belonging to the primary bodies," which suffici-
ently indicates the influence of Aristotle. Kepler first considered
gravity to be " a mutual attraction between parent bodies which
tend to unite and join together ". It is, he said, the attraction
due to the earth rather than a tendency in the stone which
causes the latter to fall.

Up to Huygens's formulation of centrifugal force it was a
real problem for the early seventeenth century scientists why
the earth and moon were not attracted into contact with each
other. Borelli, in Theorias Mediceorum plonetarum ex causis
physicis deductse (1665) supposed that there was a tendency for
the two bodies to come together, but that this was prevented by
some kind of fluid pressure. Following Kepler, Borelli believed
that the sun emanated some sort of " virtue " which kept the
planets moving in their orbits. It is a most curious fact
that although his theorems on centrifugal force were discovered
about 1659, very probably Huygens did not appreciate the im-
portance of the work in relation to the idea of a physical basis
for the solar system. The possibilities of uniting the idea of
gravitational attraction with that of centrifugal force were
realized in England by Halley and, of course, by Newton. But

i The history of the subject has been very well outlined by Duhem: La
Thtoric Physique, p. 370-414 (1906).
L 161

ifo HUYGENS's SCIENTIFIC WORK

Huygens was debarred from his due place in this most important
development by certain unfortunate preconceptions. There is
otherwise no obvious reason why he should not have forestalled
Newton by many years in regard to this part of the Principia.
The influence of Descartes here led Huygens to adhere to the
hypothesis that all change is brought about through physical
contact between bodies, either directly or through the medium
of some subtle matter which filled the intervening space.
Following Descartes he took an approach which bade to reduce
phenomena to kinetics. Besides this, however, he realized the
importance of structure or the " conformation " of physical
bodies as an explanatory principle. But change effected across
empty space seemed to him to be remote from experience and to
leave a gap in the cause-effect sequence. Using the idea of force
as the scientific mode of cause, Huygens considered it necessary
to restrict the term to the operations of bodies on each other
perhaps under the influence of Descartes 's dictum that what is
true may be clearly conceived. It was this limited view of what
can constitute a causal mechanism that led Huygens to postulate
his various " media ". For the transmission of light he required
an ether, for magnetic fields, a magnetic medium, and for gravi-
tational effects a " subtle matter ". In the Traite de la Lumicre
the situation is by no means simplified. There is a possibility, he
there remarks, " that the particles of the ether, notwithstanding
their smallness, are in turn composed of other parts, and that
their springiness consists in the very rapid movement of a subtle
matter which penetrates them from every side, and constrains
their structure to assume such a disposition as to give to this fluid
matter the most overt and easy passage possible ". Gassendi's
atomic doctrines clearly influenced Huygens. As against Boyle,
Huygens did not consider that the motion of the atoms of
ordinary matter was sufficient to explain elasticity and thermal
expansion. Such effects, he wrote, " cannot be explained without
supposing the same subtle matter in motion with an extreme
speed".

The most interesting remark on the subject of gravitational
force to be found in Huygens's writings has been quoted on page
1 1 8. In addition there are two notes in his MS. of 1668 and 1669,
which are as follows : " Gravitatem sequi quantitatem materiae
cohaerentes in quolibet corpore " and " l poids de chaque corps
suit pr6cisement la quantit de la mature qui entre dans sa com-

PLATE VI

ii

Drawing by Huygens of his Vacuum Pump of 1 668

THE CAUSE OF GRAVITY 163

position ". Huygens, then, made the distinction between mass
and weight before Newton, but these statements do not imply
that Huygens regarded gravity as something inherent in matter.

After 1661 Huygens was occupied with experiments with a
simple air pump. His interest in this instrument was aroused
during his visit to London. In 1668, an improved pump was
constructed. This machine had a brass piston bound with fine
flax in place of a wooden piston impregnated with wax. The
pump (see Plate VI) was later described by Denis Papin in
Nouvelles Experiences du Vide (1674). Actually Huygens and
Papin in their experiments of 1674-7 added little to the work of
Boyle; Huygens was not sufficient of a chemist to pursue the
more interesting problems. The observation which most
attracted interest was his discovery of the non-descent of
columns of water from inverted tubes placed in the evacuated
space of the vacuum pump receiver. Much correspondence
arose over this anomalous observation which Huygens himself
explained by assuming the existence of a subtle matter which
exerted a pressure even after the pressure due to the air had been
almost entirely removed. The effect was only noticed with water
(and later with mercury) which had been " purged of air ". As
has been noted in the earlier part of this book, Wallis doubted if
Huygens's subtle matter really explained the phenomenon: "For,
if this Matter be so subtile as to pass, through the top of the
Glass, upon the Quicksilver ... I do not see, why it should not
balance itself (above and below) in the same manner as Common
Air would do, if the Tube were pervious to it at both ends, and
the Quicksilver, by the preponderance of its own weight, fall pre-
sently." Hooke, although a Cartesian, came nearer to the truth.
In his Micrographia he mentioned the cohesion between liquids
and glass. Mercury and glass were too different in their natures
to cohere but water, " being somewhat similar to both, is, as it
were, a medium to unite both the glass and the mercury to-
gether ". In the absence of water, however, Hooke felt obliged
to accept Huygens's explanation. Even Newton allowed that
Huygens's explanation was the probable one. t Newton also sug-
gested that the rise of liquids in capillary tubes might be due
to the ethereal medium but his own adherence to the hypothesis
of an ether was, as is well known, somewhat inconsistent.

His persuasion that a subtle matter exists was a strong argu-
ment for the retention of Descartes's vortices by Huygens. In

164 HUYGENS'S SCIENTIFIC WORK

1667 he attempted to work out a satisfactory explanation of gra-
vity as an effect of circular motion. His hopes that this would
prove possible no doubt rose as the existence of a subtle matter
became more and more accepted by men of science. By 1669 he
felt himself to be in a position to put his view before the
Academic Royale, the occasion being a discussion on the subject
of gravity in which the other speakers were Roberval and Mari-
otte. In fact, no other theory was put forward and the discussion
became a criticism of Huygens's theory.

Huygens proposed to limit himself to terrestrial gravity and
for this purpose considered the Earth to be an isolated system.
The Cartesian vortex moved, according to his view, around the
Earth in such a way that the subtle matter everywhere moved
parallel to great circles on the Earth's surface. So far as one can
judge there was no question of gravity extending to the moon.
To illustrate his argument Huygens described an experiment
in which a bowl of water is rotated about its axis. Heavy particles
introduced into the rotating liquid were found to be propelled
towards the centre as the rotation slowed down. Huygens pro-
posed that the subtle matter which played such a part in his
vacuum pump experiments was in fact the matter of the vortex
about the earth. If the speed of rotation were high enough this
would account for gravity as a centripetal reaction. But it was
necessary to suppose that the circular motion of the vortex was
" natural " and not constrained. The subtle matter, Huygens cal-
culated, would have to be in rotation with a speed about seven-
teen times as great as that of the diurnal rotation of the earth.

To all this Roberval and Mariotte made the more obvious
objections, for it is difficult to conceive how an ether which is per-
meable to matter can exert a pressure on matter. Going further,
they questioned the validity of restricting all explanations to the
terms of matter and motion. They questioned the evidence for
supposing that circular motion was in this instance " natural ".
Roberval preferred the view that gravity is a mutual attraction
between the particles of bodies. He had, he said, maintained this
view as early as 1636. Both he and Mariotte considered that Huy-
gens had only replaced one mystery by another.

Huygensfs Optical Studies

The study of the propagation of light and its behaviour at
reflecting and refracting surfaces has a long history. This is, in-
deed, the oldest branch of physics and it is necessary to recall
that for centuries the subject possessed for many minds a certain
mystery which was dispelled only as the phenomena came to
be seen as illustrations of general laws. Descartes put the sub-
ject of light in a central position in his natural philosophy; one
of his works was entitled Le Monde ou Traite de la Lumiere.

Yet Descartes achieved comparatively little in his study of
light and it was Kepler's Dioptrice of 161 1 which, more than any
other single work, laid the foundations of modern optics.
Euclid's Optics (c. 300 B.C.) contained a statement of the equality
of the angles of incidence and reflection for a plane surface and
Claudius Ptolemy (c. A.D. 150) introduced the study of refraction.
Kepler spent much time on but failed to discover the relation
between the angles of incidence and refraction. His most impor-
tart relation was in fact (in modern symbols)

D

i - r

constant

Here i is the angle of incidence and r the angle of refraction and
D the angle of deviation (i - r). That is, the deviation varies as
the angle of incidence a relation which is nearly true for angles
of incidence less than 30. Kepler showed that the constant for
ordinary glass was about i. He then calculated the principal foci
in the cases illustrated :

(i) Parallel rays incident on a convex glass surface :

\

.66

HUYGENS'S SCIENTIFIC WORK

He obtained the result f = yr where f is the distance of the prin-
cipal focus F and r is the radius of curvature :

IAJ- i ' 0.5

(ii) Parallel rays incident on the inside surface of a glass block
having a convex surface:

FIG. 40

He obtained f = ar a result which follows simply from the mod-
ern relation :

/ - r ~ r
(iii) Parallel rays incident on a convex lens:

FIG. 41

He obtained the result f = r for the case of a lens having two sur-
faces of equal radii of curvature (r). This is correct, for :

= r .

Putting^ - -r 2 , f =

5 0.5 */

There were, however, no general equations for the treatment
of lenses up to the time when Huygens began his work. Cava-
lieri, following the lines of Kepler's work, in 1647 proved the rela-
tion used above for the focal length of a thin lens and Isaac

OPTICAL STUDIES 167

Barrow in 1674 found by a geometrical method the image formed
by a thick lens upon which an axial pencil falls. " Such cum-
brous geometrical investigations involving the separate consider-
ation of numerous particular cases/' writes Professor Wolf (A
History of Science, Technology and Philosophy, I, p. -248), " were
eventually superseded by the analytical methods of Descartes,
which Halley, in 1693, successfully applied to the problem of
finding the general formula of the thick lens."

Descartes did more than Kepler to treat the problem of
spherical aberration but in this and other respects he arrived at
no useful result of practical importance. His recommendations
concerning elliptical and hyperbolic surfaces were chimeras and
much needl-ess labour was lost in attempting to put his ideas into
practice.

Unfortunately Huygens's work in optics belongs to the earlier
period referred to by Professor Wolf and his writings are very
tedious to read because of the absence of algebraical formulae.
Throughout his life he was constantly amplifying and re-writing
his manuscript and only the Traitf de la Lumtire, considered in
a later chapter, appeared during his lifetime. Of the rest of
his extensive researches some were made public in lectures to the
Academic Royale but the remainder was hidden until published
in 1703 by which time it had really ceased to possess more than
historical importance.

Refractive indices are not widely quoted by Huygens in his
Dioptrica. He seems to have supposed, from the small number
of materials then available, that exact values were unimportant.
His method, in the case of glass, was to determine the focal length
of a plano-convex lens and to apply the formula

The method of finding ^ from the true depth and apparent
depth of an object seen through a rectangular block was also

FIG. 42

l68 HUYGENS'S SCIENTIFIC WORK

known to him and was used in his researches on Iceland spar.
For liquids his method was to fill a large glass cylinder and find
tfye distance of the focal line when the incident light, perpendicu-
lar to the axis, was parallel.

For a cylindrical lens we have :

Putting f=v'-r (see figure above):

.e.

or 2JJO/ - pif = i(v' - r),

u w/ - f

whence p = -- -

t; --

as given by Huygens. Huygens, however, arrived at this result
by purely geometrical methods.

Huygens's method of treating refraction may be illustrated
by the problem of finding the principal focus of a convex

FIG. 43

spherical surface. Huygens showed that if C is the centre of
curvature and NP, OB are parallel with the axis AQ,

AO
taking ~^ = (x then Q is the point through which the rays will

pass. This, of course, is correct, for, using the formula :

i* _ i __ IA- i

v ~u ~~ r

and putting u = oo, v = /,

a UL-I , -/

_!_ = L. whence JJL= L-

f ' ' *-f

OPTICAL STUDIES 169

Needless to say, no formula of the type quoted is used in the
work. But the correspondence of Huygens's geometry with
modern practice can be perceived if his diagrams are carefully
examined. His method of locating the image of a point source
produced by a lens is to employ the relation

DO DC

or

DC ~~ DP
DO.DP = DC 3

Now, DO = u-f,
i.e., uf + uf=uv,

and hence (t< - /)(w + 1;) =

or

which at once appears familiar. The original proof is, however,
too long to quote.

Huygens pointed out that there is an optical centre in a lens
such that rays passing through emerge after traversing the lens
parallel with their original directions. In Fig. 45, E and F are the

f P

FIG. 45

centres of curvature of the lens faces and ED, FB are radii. The
point L can be found since it may be shown that

BL FB

LD = ED

170 HUYGENS'S SCIENTIFIC WORK

Some consideration was given to the effect of a lens immersed
in a liquid such as water. Huygens showed how to calculate the
refractive index for the two media in contact, knowing the in-
dices with respect to air. He also gave a useful account of the
human eye. He distinguished the liquids known as the vitreous
and aqueous humours but made the mistake of supposing that
the content of the crystalline lens was also a liquid. In consider-
ing the location of an image at the least distance of distinct
vision, a fundamental idea in the theory of optical instruments,
his work was rendered unnecessarily complicated by an unfor-
tunate choice of distances. Huygens measured the distance of
the image from the eye and not from the lens through which
it was viewed. However, he was led to discover an interesting
theorem concerned with the magnification produced by a system
of lenses. This, quite briefly, stated that by interchanging the
positions of the eye and the object, without altering the posi-
tions of the lenses, the object appears to the eye to be of the same
size as before. This conclusion is of purely theoretical interest
but there is a sequel : Lagrange later obtained equations which
accord with this peculiar theorem and this in its turn led to work
by Hamilton, Clausius and Kirchhoff. There seems to be a con-
nection between Huygens's work and later developments of the
conception of optical distance.

Huygens's more important researches on lenses dealt with
the subject of spherical aberration. It was known from the time
of Kepler that the middle of a lens having spherical surfaces had
not exactly the same focal length as the peripheral pan of the
lens. To secure better definition it was customary to employ a
stop covering all but the middle of the lens. The aperture used
was judged by experience. Huygens saw that it should be pos-
sible to calculate the aperture permissible for any given lens.
It should also be possible to decide the optimum form for a
lens of given focal length. So early as 1653 Huygens compared
the distortions produced by a plano-convex lens first with the
convex and then with the plane side towards the light. He also
introduced the idea of optical thickness as measured not by the
actual thickness at the middle but by the difference in thickness
at the middle and at the edge. For a plano-convex lens the focal
length of the peripheral part can be calculated in terms of the
radius of curvature and the distance of the incident ray from
the axis. For rays close to the axis the focal length is given by

OPTICAL STUDIES 171

the ordinary formula. Huygens performed the calculation for
the two positions of the lens and showed that the separation
of the foci was less when the light was incident on the curved
face. When the curved face is turned towards the light the rays
suffer approximately equal deviations at the two refractions.
Huygens saw that spherical aberration increases with the amount
of deviation occurring. It thus appeared advantageous to use
two lenses at the eye-piece of a telescope instead of only one
and it could be secured that the total deviation was divided
equally between these two lenses. Huygens showed that this
was so when the separation of the lenses was equal to the differ-
ence of their focal lengths.

Huygens considered that the Dutch, or as it is sometimes
called, the Galilean, telescope could be made more free from
spherical aberration than the Keplerian telescope which em-
ployed a convex eye-piece. Seeing that the concave eye-piece of
the former compensated in some degree for the aberration of the
objective, a much greater aperture should be permissible and
thus a greater magnification might be obtained without increase
of length of the telescope. Huygens recognized, however, that the
Dutch telescope suffered from having too narrow a field of view
for astronomical purposes. Various means of remedying this were
considered but they were abandoned because of the difficulty
of grinding lenses to a specified form.

It was still essential, however, to define the practicable limits
to the aperture for any given lens, for there continued to be a great
deal of confusion on this point. Huygens considered definition
to depend simply on the quantity of light per unit area falling
on the retina of the eye. His procedure was to start with a tele-
scope of known dimensions which gave good results and to cal-
culate the lengths and apertures for others of the same standard
of definition. For this purpose he limited himself to a consider-
ation of the objective and ignored the eye-piece.

If f and f f are the focal lengths of two objectives and <f> and ^'
those of the corresponding eye-pieces and g and g r the linear mag-
nifications, d and df the diameters of the apertures, then, for
equal intensity of light at the eye :

d _ j _ f *L

d' ~ g> ~ ' f

** - *'* (\)

f ~ T (

172

HUYGENS'S SCIENTIFIC WORK

To compare the aberrations we may suppose, with Huygens,
that the lenses are of the " same sort ", i.e. that

where R represents the radius of curvature of a lens surface. In
Fig. 46 the distance FFj represents the separation of the foci. We
may put FF 1 = &, F'F/ = $'; then it may be shown that :

a ff ,..,

__ _ '__ (n)

9' ~ d'*f .............................. () '

FIG. 46

From the figure it is clear that the rays through the peripheral
part of the lens will meet the focal plane through F at points H
on the circumference of a circle of radius FH. FH is given by

FH=& tan 6.

The focal plane of the objective is also that of the eye-piece of an
astronomical telescope. The image of the circle will therefore be
the circle of aberration for the telescope and its radius will be
given by

8 tan
X where K is a constant.

Putting tan 6 = -7. this becomes K - . For both telescopes

2/ 2/0

to produce equal circles of aberration

ad

f

or by (ii)
and by (i)

and

(iii).

?

(iv).

TV
*/

\ J

OPTICAL STUDIES 173

These equations, derived after the manner given in the intro-
ductory notes to volume 13 of the Oeuvres Completes, summarize
the rules elucidated by Huygens.

As has been mentioned, the existence of spherical aberration
was not unknown to Huygens's contemporaries nor to some of
his predecessors. Maurolycus (in 1553) even mentioned it. New-
ton, in treating the aberration of a plano-convex lens with the
plane side towards the light found the value of the aberration,

FF l in the form of a series in which the first term was

- i

where e was the thickness of the lens and JJL the refractive index.
Picard also deserves mention for, in his Fragments de Dioptrique,

7?

he obtained a value of ~ for the spherical aberration of a glass

plano-convex lens receiving light on the convex side. His method
resembled that of Huygens. Molyneux gave a single numerical
calculation to his Dioptrica Nova (1697); he concluded that the
" depth of focus " is smallest in the case of a plano-convex lens

FIG. 47

when the rays are received on the convex surface. Huygens spent
much time on the relation between spherical aberration and the
inclination of the lens surfaces at the periphery. He saw that for
a thin lens the deviation produced at a given point may be
regarded as constant. Later he recognized that chromatic aberra-
tion was the quantity which was most affected by variation in
the angle of inclination.

When he turned to the study of chromatic aberration, Huy-
gens made use of the work of Newton. There appears to be no
evidence that Huygens measured the refractive index of glass for
different colours. Now Newton estimated the circle of chromatic
aberration (radius CO in Fig. 47) to be one-fiftieth of the diameter
of the lens employed. The fact that Huygens obtained divergent
results may have been due to the use of a very different glass or

174 HUYGENS's SCIENTIFIC WORK

to the choice of a different circle. Newton's circle was situated
midway between F v and F r ; the choice of the plane through F v
would give a circle of greater diameter.

To compare the relative magnitudes of chromatic and
spherical aberrations Huygens took a plano-convex lens of
twelve inches focal length and allowed an aperture of half an
inch. The optical thickness was thus only 1/192 inch. With the
curved surface towards the light the spherical aberration
measured along the axis is, by Picard's rule, 1/164 inch. The
chromatic aberration may be calculated from the relation
f r - f v cof where o> is the dispersive power of the glass and f is,
of course, the ordinary focal length (for the mean ray). Taking
o) to have the value .017 this means that the distance f r -fv is
i /5 inch. This is 33 times as great as the spherical aberration. In
large telescopes the difference between the effects would be even
greater. Following the method previously used, Huygens esti-
mated the aperture of any given lens by comparison with an
instrument which gave satisfactory results. He showed that the
ratio of the apertures of the lenses used for the objective and
eye-piece would be equal to that of the square roots of the focal
lengths.

When Huygens used a telescope expressly designed for observ-
ation of Saturn to study the moon he found the brightness of the
image too great. He accordingly restricted the aperture of the
objective very much but was surprised when a point was reached
when the clearness of the image suddenly diminished. He con-
sidered this was due to some property of the eye. " For also," he
wrote, " when one places in front of the naked eye a plate having
a hole of i/5th or i/6th of a line in diameter, the edges of
objects begin to appear less clear, and the confusion becomes
greater the more one diminishes the size of the aperture.' 1 Huy-
gens does not seem to have identified the effect with that of the
diffraction described by Grimaldi in his Physico-mathesis de
Lumine (1665). It is also mentioned in Newton's Principia.

Of course, certain of the conditions affecting the construction
of telescopes apply also to the construction of micioscopes. Huy-
gens was for a time a keen microscopist and he sided with Leeu-
wenhoek, Redi and Swammerdam in the view that the evidence
was all against spontaneous generation even of protozoa. Huy-
gens treated the optical system of the compound microscope in
much the same manner as he treated that of the telescope. The

OPTICAL STUDIES 1 75

diagrams he gave to explain the principal features were of more
ojf less modern form; they show the refraction of the rays so that
a real image is formed by the objective. The image, as magnified
by the eye-lens, is viewed at the least distance of distinct vision.
He derived an expression for the magnification of the instrument
and showed that if this is increased by decreasing the focal length
of the objective there is an inevitable decrease in the depth of
focus. As a practical microscopist he found that for certain observ-
ations better results were obtained when the object reflected light
than when the light was transmitted. He was responsible for
the invention of dark ground illumination.

At this point there is a temptation to introduce an account of
Huygen's excursions into biology, if only to correct the impres-
sion that he was so exclusively a student of the physical sciences.
His studies of infusoria were quite notable and in one or two
points, original, and of course the fact is that these seventeenth
century men of science took no trouble to observe any artificial
boundaries to their " subjects ". Huygens not only translated
Leeuwenhoek's writings on microscopic observations, he was a
fellow-observer who repeated and extended his compatriot *s
experiments. Like Leeuwenhoek, he was opposed to the theory
of spontaneous generation and his own experiments on the sub-
ject are a prelude to those later carried out with such perfection
by Pasteur. There is something immensely stimulating about this
all-round activity in science, this feeling of the arresting interest
of so piany and such diverse problems. But, in reading Huygens 's
correspondence and in working at his notebooks, one has to clear
away the notion of the " specialist " scientist. One sees that his
interest in the microscope is as practical as his interest in the
telescope was at all times; he wanted a science of optics in order
to make his instruments, to perform calculations in astronomy,
to understand the phenomena of " false suns " and haloes, to
pursue the questioning of Nature out into the remotest spaces of
the universe or down to the limits of the smallest living
organisms.

XI
The Wave Theory of Light

The notion that light is in nature akin to sound is very ancient.
Thinking along these lines, Roger Bacon stated that light tra-
velled in successive stages through the air; hft language hinted
at some kind of vibratory motion. After that no progress beyond
a general assumption of some analogy between light and sound
was made up to the time of Francis Bacon, and he had little
to suggest on this subject. Descartes first gave an interpretation
of the facts, so far as they were known, which had some appeal
for men of science in the seventeenth century. For him light was
to be regarded as a pressure transmitted with infinite speed
through the subtle matter which filled his universe. "Light in
luminous bodies," he wrote, " is only a certain movement, or a
very lively motion which passes towards our eyes ... in the
same way that the movement or resistance of bodies which a
blind man meets passes to his hand by the medium of his stick."
Again : "... it is not so much the movement as the inclination
to move of luminous bodies that we must consider as their light
. . . the rays of this light are nothing more than the lines along
which this inclination tends."

Huygens was not influenced exclusively by Descartes, how-
ever, and the views of Gassendi certainly deserve to be men-
tioned. The important thing about Gassendi's natural philosophy
in this connection was his admission of the vacuum as a primary
conception. He supposed that atoms of light traversed the empty
spaces between the celestial bodies and he left it open to ques-
tion whether, by analogy with sound, these atoms were emitted
periodically. Ideas obtained partly from Descartes, partly from
Gassendi, seem to have been combined by Huygens but there
was much that was new about his own theory. As has been men-
tioned, Hooke, in his Micrographia of 1665, spoke in general
terms of waves of light propagated with finite speed. Grimaldi,
in his Phy$ico*nvtkesis de Lumine of the same year, pursued
similar ideas; there is in this work a figure which may represent
light as propagated by transverse vibrations. He spoke of the
motion of a fluid medium as being spiral in form. Whether Huy-

176

THE WAVE THEORY OF LIGHT 177

gens read this work is uncertain but it was amongst his books
at the date of his death.

Perhaps more importance should be attached to the work of
the Jesuit I. G. Pardies (d. 1673), f r t ^^ s amateur student of
science showed the greatest faith in the Analogy between light
and sound. It is known that he showed his completed work to
Huygens. The latter never accepted so close an analogy but one
point of resemblance has been remarked: Pardies supposed that
the pulses by which light was propagated were irregular in period.
The work Optique, published in 1682 by Ango, contains some of
his ideas. The first to develop the idea of periodicity in the pro-
pagation of light was Newton (1672), and Malebranche was the
first French writer to follow him (1699). Huygens seems to have
been led to make an independent study of the problem as a
result of the perplexing properties of Iceland spar; it is known
that he conceived the idea of spheroidal waves within this crystal
while at the Hague in 1677. There is little to be gained, however,
by attempting to maintain that Huygens originated the wave
theory de nova. He may well have obtained the initial idea from
Pardies's manuscript. His great achievement lay in presenting the
theory in a form in which it could be fruitfully applied through
the development of a suitable geometry.

Huygens did not invoke a Cartesian medium for the propa-
gation of light until after 1668, when it seemed to him that he
had definite evidence from his vacuum pump experiments for the
existence of such a medium. He then sought to combine this
hypothesis with an atomic theory similar to that propounded by
Boyle. He agreed with the latter that liquids and solids alike are
composed of particles in proximity but he supposed that there
were intervening spaces and that these were filled with the much
smaller particles of a subtle matter. The elasticity of the air
demonstrated by Boyle's experiments "seems to prove," he wrote,
that it is made up of particles floating and " agitated very rapidly
in the ethereal matter composed of much smaller parts ". Huy-
gens's atomism is peculiar therefore in that he extended it to
subtle as well as to ponderable forms of matter. That a consider-
able proportion even of solid bodies is occupied by the ethereal
medium is clear, he considered, from the fact that so dense a sub-
stance as gold does not screen off the effect of a magnet or of
gravitation from a body.

Huygens considered his kinetic theory of matter was only a

M

178 HUYGENS'S SCIENTIFIC WORK

beginning. He was forced to introduce " soft " particles to damp
out the motion of light in opaque bodies and his speculations
led him in one place to speak of the particles of ether being com-
posed of still smaller parts and penetrated by a second subtle
matter. There is some inconsistency in his statements and we are
left in some doubt whether he always intended a clear distinction
between a luminiferous ether on the one hand and subtle media
for the propagation of gravitational and other effects on the
other. He did distinguish an ether and a matiere subtile but the
grounds of the distinction are not really clear. The nearest Huy-
gens came to simplifying his ideas was to suggest that the media
concerned in the propagation of light and in the anomalous
vacuum pump experiments are the same. He explained that he
could not identify the luminiferous medium with that which
causes gravity since the latter was in his view found only near the
earth. The weakness of his position here has already been dis-
cussed. As Clerk Maxwell remarked, " To fill all space with a
new medium whenever any new phenomenon is to be explained
is by no means philosophical."

The Traite 1 begins with an admission that some of the sug-
gestions are only hypothetical : " whereas the Geometers prove
their Propositions by fixed and incontestable Principles, here
the Principles are verified by the conclusions to be drawn from
them; the nature of these things not allowing of this being done
otherwise. It is always possible to attain thereby to a degree of
probability which very often is scarcely less than complete
proof." It is in keeping with this view of scientific method
to start with the hypothesis that light is a form of vibratory
motion propagated in spherical waves or surfaces as in the pro-
pagation of sound. The chief differences between the two cases
were the incomparably greater speed of light estimated by Roe-
mer and the media through which the vibrations travel. It was
also obvious that the particles of a luminous body vibrate inde-
pendently of each other and that the frequency of vibration (if
one can use such a term in relation to Huygens's ideas) is very
much higher than in the case of sound. Huygens then proceeded
to explain how light could be conceived as a succession of com-
pressional or longitudinal vibrations passing through contiguous
ether particles. Contrary to the corpuscular theory there was
no movement of translation. This explained how two light rays
1 Traite de la Lumi&re.

THE WAVE THEORY OF LIGHT 179

could travel in opposite directions in the same space, or cross at
an angle, without hindering each other. It also led to Huygens's
famous conception of secondary wavelets, for each particle in the
path of a disturbance was a centre from which the disturbance
spread outwards through all the particles in contact. Huygens
saw that there was a difficulty here in that loss of impulses later-
ally must weaken a ray as it proceeds. The limits he set to the
" wave front " or common tangent appear quite arbitrary. As
is well known, Newton was not satisfied with this discrepancy;
it was only much later that wave-spreading in the form of diffrac-
tion effects was fully demonstrated.

Huygens's principle is often considered to be open to the
objection that a wave-front travelling backwards towards the
source can be constructed in theory but is never observed in
practice. The answer to this objection is that Huygens did not
evolve his principle from geometry so much as from his study of
elastic collision. If the particles of a medium are all equal, any
impulse received at A will be transmitted through the train of
particles until any given particle C is moved in the same direc-
tion as D. After collision with the next stationary particle E there
will be a rebound only if C were smaller in mass than E. Huy-

o ccooo o

DA C E

FIG. 48

gens supposed all the particles of the ether to be of the same
size but allowed that the effect would exist if smaller particles
were present. He doubted if such a back-wave would generate
the sensation of light. There is, however, something very un-
satisfactory in considering a point of a wave as the centre of
another wave.

The construction for regular reflection at a plane surface is
sufficiently well known to require only brief notice here. It is
given in text books of optics. Instead of progressing towards
GMMMMB the secondary wavelets can radiate outwards only
above the reflecting surface AB. While a secondary wavelet from
C is travelling to B a wavelet starting from A must have travelled
through a distance AN equal to CB. The radii of intervening
wavelets are determined in the same way and thus the common

l8o HUYGENS's SCIENTIFIC WORK

J G

FIG. 49

tangents BN can be found. This " terminates the movement "
and is the new wave-front. By geometry it may readily be shown
that, as a consequence of this construction, the angles of incid-
ence and reflection are equal.

The validity of Huygens's principle here depended, as its
author saw, on the constancy of the speed of light during reflec-
tion. Without naming the source he referred to Hooke's law in
answering this question : " This [constancy of speed] comes
about from the property of bodies which act as springs, of
which we have spoken above; namely that whether compressed
little or much they recoil in equal times/'

It was otherwise when the passage of light from one medium
to another was considered. An important difference and one
which became decisive between the wave theory and the cor-
puscular theory was Huygens's conclusion that light must travel
more slowly in the denser medium. This was not necessarily be-
cause the medium was different : Huygens considered the ethe-
real medium penetrated all solids and liquids. The difference in
speed resulted from the detours of the waves around the more
solid particles of the elements. If we ask how Huygens knew
that the ether penetrates substances his answer, given in the
Traitf, adduces the entry of ether into the Torricellean space and
that hollow bodies possess an inertia which is in strict proportion
to the mass. Like Newton, he could detect no friction of solids

THE WAVE THEORY OF LIGHT

181

with the ether. In some substances, however, Huygens supposed
the material particles were not unaffected by the light vibrations.
In this case they transmitted the vibrations also and the existence
of this second mode explained double refraction.

In his treatment of ordinary refraction, Huygens supposed a
wave-front AC (Fig. 50) to impinge on the surface of a second
medium. The speed of light in this second medium (glass) was
supposed to be two-thirds of its value in air. The radius of the
wavelet from A is then two-thirds of the distance which the
wavelet would travel in air in the same time interval. It follows
that while a wavelet from C is travelling to B, a wavelet from A

G

FIG. 50

will spread into the new medium through a distance AN equal to
two-thirds of CB. It is clear in each case that the radius of the
wavelet from K will be two-thirds of the distance KM which
the wavelet would have travelled in air. The new wave-front is
the common tangent NB. It follows by geometry that

sin L DAE velocity in air

sin L NA F ~~ ^ "" velocity in glass

When light goes from glass into air the ratio of the speeds
is inverted. In the second figure (Fig. 51), therefore, /!N=3/a BC
or^l^AG. Here we have

sin L DAE z

sin L NAF = "3"""
For larger angles of incidence (DAE) it is clear that when

1 82

HUYGENS's SCIENTIFIC WORK

But

sin LDAE =

sin L NAF =
sin Z DAE

BC
AB

NA
AJB

2

sin /.'NAF ~ 3"

sin L NAF = i and the angle of refraction be-
comes 90. Beyond this limiting condition, as Huygens puts it,
the wave-front BN " cannot be found anywhere, neither conse-
quently can A N . . . thus the incident ray DA does not pierce the
surface AB ".

F

a

B

FIG. 51

It was impossible to give a satisfactory physical reason for
the failure of the ray to penetrate the surface when the angle of
incidence exceeded the critical value. The interior reflection
which occurred took place, he supposed, " against the particles
of the air or others mingled with the ethereal particles and larger
than they ". He could not explain how the reflection could take
place when the air was replaced by a vacuum.

At the end of his chapter on refraction, Huygens showed that
his principle of secondary wavelets was in conformity with Fer-
mat's principle that the path actually taken by a ray of light
in passing between two points is the path of least time. Fermat
was in strong opposition to Descartes, whose false " demonstra-
tion " of Snell's law of refraction required that the velocity of
light be greater in a dense medium than in air. In a long letter
to de la Chambre, Fermat explained that it was necessary to
make the opposite assumption. He showed how the sine law
of refraction could be deduced from the least-time principle and
a copy of the letter was sent to Huygens in 1662. Huygens

THE WAVE THEORY OF LIGHT 183

was at first scornful of Fermat's principle, which seemed to him
to savour of Aristotelianism. This " pitiahle axiom " was one
which he had never seen usefully applied, he remarked. Never-
theless, he changed his mind over Fermat's principle and was de-
lighted when he succeeded in deriving it by his own methods.

The relation between Fermat's principle and Huygens's con-
struction in this case may be summed up by the statement that
both give the same physical interpretation of the refractive
index. The significance of this important theorem was not lost
on Huygens, though it is to be doubted if he ever imagined its
future development. There is, however, an interesting applica-
tion of the least-time principle in the Traite to atmospheric
refraction, a subject of obvious importance to astronomers.
Huygens pointed out that spherical wavelets would only be set
up in a medium which was homogeneous, or to use the modern
term, isotropic. In an anisotropic medium, as will be seen,
Huygens showed that the wavelets might be ellipsoidal in form.
The problem of the atmosphere was somewhat different. There
was here a gradual change of refractive index with density, and
Huygens saw that the wavelets must have surfaces of equal time
from the source. To include the effect of density in his wave
theory it was necessary to assume either that gaseous particles
acted as a hindrance to the vibrations or else that they trans-
mitted light themselves but did this inefficiently.

Huygens himself foresaw that later workers would furnish
what was needed to complete his " imperfect knowledge/' that
much yet remained to be done to make a satisfactory theory.
The remainder of the Traite was taken up with a striking
attempt to extend the wave theory to the phenomenon of double
refraction. As is well known, Huygens elucidated the nature of
the wave surface for the extraordinary ray as an ellipsoid of
revolution and this great achievement still stands. What
Huygens could not explain (since his " waves " were longitu-
dinal ones) was the effect of superimposing two calcite crystals in
different positions. He succeeded, that is, in working out a con-
struction for the extraordinary ray on the basis of an ellipsoidal
wave theory, but could not on these grounds account for what we
now term the polarization of the transmitted light.

The properties of calcite are described in some of the larger
works on optics, notably Mach's The Principles 'of Physical
Optics. The essential property is that, in general, an incident

i8 4

HUYGENS'S SCIENTIFIC WORK

ray, on entering the crystal, is split up into two rays, one of
which (the ordinary ray) is refracted according to the usual laws
of refraction, while the other (extraordinary ray) is not. The
substance was, Huygens admitted, anomalous and its behaviour
" seemed to overturn our preceding explanation of regular re-
fraction ". From the first, however, Huygens, on account of his
atomic theory, inclined to the view that the optical properties
of crystalline solids were to be correlated with their fine structure.
He therefore began his account in the Traite with a description
of the geometry of the Iceland spar crystal.

The crystal (Fig. 52) has the form of an oblique parallel-
epiped and there are cleavage planes in three directions parallel
with the three pairs of parallel faces. The angles of the parallelo-
gram sides were given by Huygens as 101 52' and 78 8'. The
crystal has two opposite corners which are formed by three
obtuse angles while the others are formed each by two acute and
one obtuse angles. If the obtuse angle ACB of the parallelo-
gram face at the blunt corner C is bisected by the line CE and a
plane is imagined to pass through CE perpendicular to the

D E

FIG. 52

parallelogram face, the plane also contains the edge CF. The
plane thus determined, and any other plane parallel to it, was
termed by Huygens a principal section.

As Bartholinus had shown, every incident ray, with certain
exceptions, gave rise to two refracted rays, one of which
was normal in its behaviour and exactly comparable with the
refracted ray in any ordinary medium. The second ray showed,
in general, marked abnormalities. So long as the plane of incid-
ence coincides with a principal section HH (Fig. 53), bath of the

THE WAVE THEORY OF LIGHT 185

refracted rays remain in this plane. For other planes of in-
cidence the extraordinary ray is formed in a different plane.
Moreover, while a ray S incident normally and an oblique ray
R making a certain angle of incidence, both in the plane of a
principal section, are refracted in the normal manner (rays
marked O), the extraordinary ray formed exhibits a peculiarity.
This is that a ray incident normally in the principal section
gives an extraordinary ray e which is deviated by 6 40' towards
the blunt corner C; on the other hand, a ray R, incident in the
principal section at 73 20' (almost parallel with the edge CF in
the first figure) gives an undeviated extraordinary ray e.

By a method which was in effect the measurement of the
true and apparent depth of a small object seen through the
crystal, Huygens found that the refractive index for ordinary
rays in the principal section (or any other plane) was constant
and approximately 5/3. Using a similar procedure to find
the refractive index for extraordinary rays it was obvious that
this was not a constant. The apparent depth of a point source of
light varied with the orientation of the crystal. Nevertheless,
Huygens discovered one important rule for the extraordinary
refraction which may be explained as follows. In Fig. 54, the
parallelogram GCFH is the principal section.

IK is a ray normal to the surface and KM is the extraordinary
ray. Huygens found that for rays VK, SK, making equal angles
on either side of the normal IK, the extraordinary refracted rays
KX and KT make MX and MT in HF equal. This may be called
Huygens's rule for the extraordinary refraction in this plane.

i86

HUYGENS'S SCIENTIFIC WORK

These facts had, if possible, to be collated in a single theory
of transmission. The ordinary refraction offered no difficulties.
For this the theory of spherical wavelets spreading with a speed
less than that in air was adequate. " As to the other emanation
which should produce the irregular refraction," Huygens wrote,
" I wished to try what elliptical waves, or rather spheroidal
waves would do ... ". These perhaps spread indifferently in
both ethereal and material particles and in the regular arrange-
ment of the particles might lie the source of the spheroidal wave
form : " I scarcely doubted that there were in this crystal such an
arrangement of equal and similar particles because of its figure
and of its angles with their determinate and invariable
measure ".

Working on this assumption, Huygens's construction for the
extraordinary ray from normal incident light was as follows. In
Fig. 55, RC is a wave front and AB is the surface of the crystal.
The plane is that of the principal section. Hemispheroidal
waves originate at AKkkB. The axes or major diameters of
these are oblique to the plane of AB as shown by AV: "I say
axis or major diameter because the same ellipse SVT may be
considered as the section of a spheroid of which the axis is AZ
perpendicular to AV" wrote Huygens. For the present he
considers only sections of the spheroid which are elliptical in the
given plane of the figure. The common tangent to the semi-
ellipses is NQ; and this is the propagation of RC as in Huygens's
original theory. NQ is parallel with AB, but is displaced later-
ally as required by the refraction of the extraordinary ray.

R H h h C

Fic - 55

THE WAVE THEORY OF LIGHT 187

It was next necessary to find the exact form of the ellipsoid of
revolution and the orientation of axes in the crystal l . Fortun-
ately all six faces of the parallelepiped produce the same refrac-
tions : the substance was uniaxial. Picturing the appearance of
the blunt corner of a calcite rhomb (Fig. 56), and imagining the
three principal sections respectively normal to each of the three
faces, these intersect in a line, called by Huygens the axis of the
corner, subtending equal angles with each of the three edges to
the corner. If, now, the direction of the axis of the wave
spheroid of rotation of which Huygens first thought did not
coincide with that of the axis of the corner, each of the three
principal sections would not be characterized by the same optical

FIG. 56 FIG. 57

properties. The inclination of the axis of the corner to each of
the faces of the corner amounts to 45 20'. The orientation of
the spheroid being known, the fact that for normal incidence the
extraordinary ray is deviated in the principal section from the
ordinary ray by 6 40' towards the blunt corner is sufficient to
establish the shape of the spheroid. By calculations based on his
data, Huygens found the following to be in agreement with the
facts. If OA (in Fig. 57) is the axial direction of the calcite and
the ordinary wave spreading out from the point O of the crystal
is represented by a sphere of radius OA, the surrounding oblate
spheroid of rotation AB with axis of rotation OA represents the
corresponding extraordinary wave emerging simultaneously
from O. The ratio of OA to OB is as 8 to 9 (very nearly) while the
ratio of OA to the corresponding path in air is as 3 to 5.

The construction for the refracted rays from oblique incident
light may be explained as follows. The plane of incidence is con-

1 I am indebted for this passage to the excellent summary of Huygens's
Ttaitd in Mach's Principle* of Physical Optics.

l88 HUYGENS'S SCIENTIFIC WORK

sidered to lie in the plane of the paper; MN is the calcite-air sur-
face (Fig. 58) and SO an incident ray.

Let SO bfc produced to S' and let a sphere of any convenient
radius be described about O as centre. At its intersection with
SOS 7 let a tangent plane be constructed intersecting MN in Q. If
now a sphere of three-fifths the radius of the former is described

M \0 Q N

E

about O, its point of contact R with the tangent plane through
Q gives the ordinary ray OR. Let OA be the direction of the
axis of the calcite rhomb. A spheroid is now described about the
smaller sphere such that its axis of rotation OA (the minor
axis), which equals OR, is 8/9 the length of the major axis.
The point of contact T of the tangent plane through Q to the
spheroid then gives the extraordinary ray OT. This construction
is confined to one plane only when the axis is symmetrical with
respect to the plane of incidence, that is, either coincident with
it or perpendicular to it. For any other orientation the extra-
ordinary ray is inclined to the plane of incidence. The results
were studied experimentally by Huygens, who cut the crystal
so that the optic axis was normal to the surface, parallel with the
surface and in the plane of incidence, among other forms. He
also found experimentally that for rays in the principal section
the extraordinary ray was not refracted when the angle of in-
cidence was 1 6 40'. What he had to do here was to show that
for this angle the rays continue without refraction since they
are directed along the major axis of the ellipsoid. The fact that
there is no bending in spite of the change in velocity in the new
medium is of course explained by the fact that the new wave

THE WAVE THEORY OF LIGHT 189

front is not normal to the direction of the rays as is the case for
spherical wavelets.

On the basis of his theory, Huygens clearly expected that it
would in general be possible to split up rays which emerged from
one crystal of calcite by passing them through a second crystal.
Excluding certain special positions of the crystals which he him-
self understood, it would be expected that the ordinary and
extraordinary rays would be split up again on entering a second
crystal. This was found to be by no means the case. Huygens
was considerably perturbed by the discovery and laboured hard
to explain it, but in vain. He was, as Mach remarks, on the
threshold of a great discovery the transverse nature of light
waves but his conceptions hindered his taking this step for-
ward. "Before finishing the 'treatise on this crystal/' he wrote,
" I will add one more marvellous phenomenon which I dis-
covered after having written all the foregoing. For though I have
not been able till now to find its cause, I do not for that reason
wish to desist from describing it, in order to give opportunity to
others to investigate it. It seems that it will be necessary to make
still further suppositions besides those which I have made; but
these will for all that not cease to keep their probability after

\A /A

having been confirmed by so many tests." In this Huygens was
correct, his geometrical analysis of the ellipsoidal wave still
stands. He went on (Fig. 59) : " The phenomenon is, that taking
two pieces of this crystal and applying them one over the other,
or rather holding them with a space between the two, if all the
sides of one are parallel to those of the other, then a ray of light,

190 HUYGENS's SCIENTIFIC WORK

such as AB, is divided into two in the first piece, namely, BD
and BC, following the two refractions, regular and irregular. On
penetrating thence into the other piece each ray will pass there
without further dividing itself in two; but that one which under-
went the regular refraction, as here DG, will undergo again only
a regular refraction at GH; and the other, CE, an irregular re-
fraction at EF. And the same thing occurs not only in this dis-
position, but also in all those cases in which the principal section
of each of the pieces is situated in one and the same plane, with-
out it being needful for the two neighbouring surfaces to be
parallel."

In these words Huygens described his discovery of the
polarization of light. He went on: " Now it is marvellous why
the rays CE and DG, incident from the air on the lower crystal,
do not divide themselves the same as the first ray AB. One would
say that it must be that the ray DG in passing through the upper
piece has lost something which is necessary to move the matter
which serves for irregular refraction." "... It seems that one is
obliged to conclude that the waves of light, after having passed
through the first crystal, acquired a certain form or disposition in
virtue of which, when meeting the texture of the second crystal,
in certain positions, they can move the two different kinds of
matter which serve for the two species of refraction; and when
meeting the second crystal in another position are able to move
only one of these kinds of matter. But to tell how this occurs, I
have hitherto found nothing which satisfies me/' The twenty-
sixth query at the end of Newton's Optics referred to this
problem. Has not a ray of light two sides, Newton asked, and
his question became one of extreme significance after Young
made the suggestion that the wave motion of light is not
longitudinal but transverse.

Huygens did not in the Traite de la Lumierc attempt a
detailed physical explanation of the production of a spheroidal
wave but he communicated on this subject with Papin. In a
letter written in December 1690, he wrote: "As to the kinds of
matter contained in Iceland crystal, I suppose one composed of
small spheroids, and another which occupies the interstices
around these spheroids, and which serves to bind them together.
Besides these, there is the matter of ether permeating all the
crystal, both between and within the parcels of the two kinds
of matter just mentioned; for I suppose both the little spheroids,

THE WAVE THEORY OF LIGHT \g\

and the matter which occupies the intervals around them, to be
composed of small fixed particles, amongst which are diffused
in perpetual motion the still finer particles of ether. There is
now no reason why the ordinary ray in the crystal should not be
due to waves propagated in this ethereal matter. To account for
the extraordinary refraction, I conceive another kind of waves
which have for vehicle both the ethereal matter and the two
other kinds of matter constituting the crystal. Of these latter, I
suppose that the matter of the small spheroids transmits the
waves a little more quickly than the ethereal matter, while that
around the spheroids transmits these waves a little more slowly
than the same ethereal matter . . . These same waves, when
they travel in the direction of the breadth of the spheroids, meet
with more of the matter of the spheroids, or at least pass with
less obstruction, and so are propagated a little more quickly in
this sense than in the other; thus the light-disturbance is propa-
gated as a spheroidal sheet/' Papin and Huygens did not, how-
ever, see eye to eye over atomic doctrines and Huygens was
critical of Papin's more or less orthodox Cartesian ideas.

Professor Sylvanus Thompson, in his " Note by the Trans-
lator " which prefaces the English edition of Huygens 's Traite,
remarked : " If Huygens had no conception of transverse vibra-
tions, of the principle of interference, or of the existence of the
ordered sequence of waves in trains, he nevertheless attained to a
remarkably clear understanding of the principles of wave-
propagation; and his exposition of the subject marks an epoch in
the treatment of optical problems." There seems to be fairly
general agreement that Huygens's theory ought more properly
to be called a pulse theory. The concept of periodicity was no-
where introduced. Some evidence, though not of very substan-
tial character, can be found in Huygens's note-books to show
that he did speculate that waves (ondes) exist. He did not show
much interest in the problem of accounting for colour, and
although he made some favourable comments on Hooke's ex-
planation of colour, he himself felt the matter lay outside mathe-
matical treatment. Newton first introduced the idea that colour
is related with frequency, but he received scant encouragement
from Huygens. It is curious that Newton, the founder of the
corpuscular theory, at least in his later speculations, came nearer
to a wave theory of the modern type than Huygens did. He
never threw over the corpusular theory, however, on the grounds

192 HUYGENS'S SCIENTIFIC WORK

that first, a wave motion should show some lateral spreading (not
yet identified with Grimaldi's discovery of diffraction) and
second, that no waves as then conceived could produce the effects
discovered by Huygens when light passed successively through
two crystals of Iceland spar. As Whittaker remarked, " his
objections are perfectly valid against the wave theory as it was
understood by his contemporaries, though not against the theory
which was put forward a century later by Young and Fresnel ".
The idea of transverse vibrations was, in Whittaker's opinion,
narrowly missed by John Bernoulli the younger, whose prize
essay on the ether was written in 1736.

The very last part of Huygens's Traitt is an echo of
Descartes. Here Huygens wrote on " the Figures of the trans-
parent Bodies which serve for Refraction and Reflection ".
Here Huygens treated problems of aplanatic surfaces by means
of his wave construction and by employing the least-time prin-
ciple. In this way he gave proofs for some of the propositions
Descartes had merely stated and for which his method had not
been recorded. Huygens acknowledged the importance of conic
sections. He proved that concavo-convex lenses of the form
described by Descartes would be aplanatic, and determined the
conjugate foci. Following Descartes he proposed that the concave
surface of such a lens should be spherical. The impression given
by the work, however, is that its interest is from the start purely
theoretical. Huygens certainly recognized that it was of little
use to demonstrate that refracting surfaces of unlimited aperture
could be aplanatic so long as chromatic aberration was without
remedy- His constructions for reflection from curved surfaces
were, however, of permanent value and his general method of
finding the focus from a consideration of wave fronts has proved
most useful.

There is no question that Huygens's principle of the propa-
gation of light by secondary wavelets has been of great import-
ance in optics. Many of the questions left unanswered by
Huygens were settled by Fresnel. But the early history of
Huygens's wave theory was disappointing. It is true that the
Traite de la Lumtere was well received. The wave theory was
even expounded at the University of Wittenberg in 1693. Soon
after Huygens's death, however, the theory seems to have fallen
into general neglect. In Holland neither s'Gravesande nor
Musschfenbroek adopted it.

XII

Saturn

As has been described in the notes on Huygens's life, the
planet Saturn presented a puzzling appearance when viewed
through the first telescopes. Jupiter's satellites were a spectacle
which could not but strengthen belief in the principles enunci-
ated by Copernicus, but this was far from being the case with
Saturn. Galileo ended his studies of Saturn with the words
altissimum planetam tergeminum observavi, but concealed in the
form of an anagram : " I have observed the most distant planet
to be triform." A system of three bodies, two of them rotating
about the middle one would, however, be a sufficiently curious
spectacle to arouse further attention.

Some account of the contents of Huygens's Systema has been
given in the early part of this book. The work of observing the
planet, when the weather was favourable, occupied Huygens
during the years 1655-9, and this involved a considerable
amount of work in the construction of telescopes. The rings
were fortunately becoming more inclined during this period, and
in 1664, five years after his book had appeared, Huygens drew the
rings at a great inclination. He was, of course, mistaken in sup
posing a single solid ring. The Cassini division is visible in a small
telescope only when the rings appear fully open, a condition
which was not fulfilled during Huygens's earlier studies. In 1675,
however, he made a drawing which suggests that he abandoned
the view that the ring is solid (page 194).

In regard to the sidereal period of the planet and the constant
inclination of the rings to the ecliptic Huygens was more or less
correct. He remarked in his preface on the equal inclination of
the axes of Saturn and the Earth to the ecliptic, and he made
it abundantly clear that the book was essentially a comment on
Copernicanism. He accordingly did not consider it out of place
to mention his search for moons of Mars, Venus and Mercury.
In observing the four known moons of Jupiter, he remarked a
variation in the appearance of the disc of the planet which he
ascribed to the presence of vapours comparable with our clouds.
Less related with his main subject, but worthy of record, was his
N 193

94

HL'YCENS's SCIENTIFIC WORK

I. p. 322. One of the
first drawings.

II. p. 224. Letter of Sept
24, 1658.

V. p. 109. In letter to
Moray, Aug. 29, 1664.

VI. p. 443. Saturn
observed May, 1669
(Huygens, Picard and
Cassini).

A drawing from
Huygens's Manuscript
K. (1675).

Dec. 27, 1657.

Drawings of Saturn after those made by Huygens
The references are to the volumes of the Oeuvres Computes.

SATURN 195

description of a nebula in Orion. This nebula, discovered by
Cysat in 1619, he considered to be essentially different from the
Milky Way, in that it did not lose its nebulosity when viewed
through a telescope. He suggested that the light came from a
more distant region visible through a hole in the black sky.
Much later (1733), Derham questioned if the nebulae "in all
Probability, be Chasms, or Openings into an immense Region of
Light, beyond the Fix'd Stars."

In regard to Saturn, Huygens's method was to collect together
all drawings of the planet then published and show that nearly all
could be explained on the hypothesis of an imperfectly visible
ring. From the outset, as he himself admitted, it had always
seemed obvious that the planets were analogous with the Earth;
consequently it is probable that all turn on their axes. A rather
more dangerous proposition was that all the matter between
Saturn and its moon (Titan) turns about the planet with periods
increasing with the radius. But it seemed obvious that the dis-
position of rotating matter was at any rate symmetrical, for the
appearance of the planet changed only slowly. Huygens found
that the plane of the ring was inclined at about ao to the
ecliptic and that this angle remained constant as in the case of
the inclination of the Earth's equator. He could not accept any
suggestion that the ring was merely an exhalation and an
evanescent phenomenon. The solid and permanent nature of
the ring could, he said, be clearly perceived. The figure he gave
in the Systema (Fig. 60) left no room for doubt as to his own
view on this point. But such a ring was certain to appear

FIG. 60

eminently unstable unless one accepted its equilibrium under
central gravitational forces. This Huygens proposed was the
true explanation. He did not suggest that the gravitational force
on the elements of the ring was balanced by the centrifugal
force due to rotation and thus did not anticipate Newton's

196 HUYGENS's SCIENTIFIC WORK

theory. His conception was a static one and required that the
ring be mechanically able to withstand the gravitational effect
without fracture. Rotation then did not come into the question.
But the idea that Saturn's gravity extended to the ring may well
have been an important advance, and may even have started
Newton on some interesting speculations.

Having disposed of effects due to aberration and obscurity of
early telescopes, Huygens was faced with the necessity of show-
ing how the hypothesis of a ring inclined to the ecliptic pro-
vided an explanation of all the various appearances of the
planet. With the telescopes used at the time the ring appeared
invisible when there was anything up to 2 between the line of
sight and the plane of the ring. The problem of forecasting the
future appearances was a purely geometrical one. In Fig. 61,
ANC represents the orbit of Saturn, DEF that of the Earth, and
L the position of the Sun. The orbits are regarded as circular
and lie in the same plane. Since the inclinations of the axes of

FIG. 6 1

the Earth and of Saturn are parallel, the equinoctial line AC
passes through the equinoctial points for both planets. Now
since the axis of Saturn remains parallel with itself the line
of intersection of the planes of the ring and the orbit is always
parallel with the line AC. If Saturn is at H, and the Earth at D,
and HM is the line of intersection of the planes of the ring and
orbit, if the angular displacement of the Earth from the line AC

SATURN 197

is greater than that of Saturn, the line HM will fall between the
Sun and the Earth. The plane of the ring passes between L and
D and no reflected light from the ring can reach D. On the
contrary, when the Earth is at less angular displacement than
Saturn, as in the relative positions N and F, or on opposite sides
of CA, as at N and f, the same surface of the ring would be
visible from either F or f.

Knowing the synodic period of Saturn, that is the interval
between two successive oppositions of the planet, it was possible
to calculate the dates of reappearance of the round form.
Huygens showed that the ring would appear thin from April to
June 1671, and vanish from sight in July or August. Not until
July or August 1672 (after an heliacal rising and setting) would
the ring appear again. It would then remain visible until 1685.
In this year, and again in 1 700, the planet would appear in the
round form. Actually, Huygens had to recognize errors in these
predictions since events were slightly ahead of schedule even in
1671. Huygens did not make very exhaustive observations, and
it appears likely that his fundamental determinations were not
sufficiently refined. He did not attach great importance to long-
continued and uninterrupted observation. Indeed, many of his
observations passed unrecorded.

The telescope first used in the observation of Saturn had a
magnification of about fifty. The objective was plano-convex,
and the eyepiece was a simple lens of about eight centimetres
focal length. An objective answering to Huygens's description
of it, and dated February 3rd, 1655, was discovered at Utrecht
University in 1867 by Harting (who wrote a short biography of
Huygens in Dutch). Huygens's second telescope was one of 23
feet about twice as long and twice as powerful as the first.
This was used after February 19, 1656. The method of com-
puting the magnification was either by determination of the
ratio of the focal lengths of the objective and the eyepiece or by
comparison of the angular dimensions of a distant object seen
through the telescope and by direct vision.

In his work on Saturn, Huygens used a micrometer consisting
of a lamina of brass. This form, which has already been men-
tioned, was used up to the end of 1659. In 1666, he adopted the
use of cross wires arranged in squares. The movable thread
micrometer invented by Gascoigne superseded this. The values
for the diameters measured by Huygens were all too large, but

198 HUYGENS'S SCIENTIFIC WORK

they were an improvement on those given by Riccioli. Also
they bear a very fair relation among themselves. As is shown
in the table, Huygens's ratios foil the diameters of the planets
compared with the Sun are all in the direction of over-estimating
the planetary diameters.

Planet. Ratio of diameter to True values,

that of Sun.

Venus i

Mars i

Jupiter i

Saturn's ring i

84 i : 112

1 66 i : 202

5-5 ' : 9-8

7.4 i : 1 1.6

While studying the apparent diameters of Jupiter and Mars,
Huygens noted the existence of bands or zones across these
planets. Drawings were given in the Systema Saturnium.

Huygens adopted Copernicus's proportions for the planetary
distances from the sun. There was, until 1672, no agreement
among astronomers over the distance of the earth from the sun.
It was therefore necessary to proceed on a probable estimate if
such could be found. Huygens's method was, having observed
the apparent diameters given above, to adopt for the Earth a size
which agreed best " with the order and good disposition of the
whole system ". In a way which reminds one of Kepler, this dis-
position seemed to Huygens to rest on a proportionality between
size and distance from the sun (Jupiter and Saturn being excep-
tional). Thus the earth, being intermediate between Mars and
Venus, probably possessed a volume intermediate between these
planets. From the figures given above, the mean (of 1/166 and
1 784) is i / 1 1 1 . Huygens took the estimate that the sun's diameter
was i / 1 13 of its mean distance from the earth. This gave the dia-
meter of the earth as i / 12543 of the sun's mean distance. The
maximum and minimum distances of Saturn came out at
122000 and 100344 terrestrial diameters respectively. A modern
estimate would put the result at about 123600 and 100200 respect-
ively, so that, considering the precarious basis of Huygens's
calculation, the result was much better than might have been
expected.

In his account of his method of observing apparent diameters
Huygens explained that he used a diaphragm at the focus of the
objective. This diaphragm had a hole a little smaller than the
diameter of the eye-piece and in this way a sharp edge to the field

SATURN 199

of view was obtained. It was easy to find the angular size of
the field of view by timing the passage of a star across it, using
the pendulum clock " recently invented ". His own telescope (pre-
sumably the one of twenty-three feet) embraced a field of o if
15". The description of a micrometer consisting of copper rods
of diminishing diameters which could be inserted in the focal
plane of the eye-piece ends the Sy sterna.

Huygens made further observations of Saturn in later years
and redetermined, among other things, the ratio of the dia-
meters of the ring and the planet. On July 16, 1667, Huygens
determined, probably with Buot, the hour at which the ring ap-
peared parallel with the horizon at Paris. This enabled him to
calculate the inclination of the ring to the ecliptic, but the details
of the work are lacking. The most important work on Saturn in
later years was done by Cassini, working at the Paris observatory,
Cassini discovered a second satellite (October 1671) and observed
the division in the ring which now bears his name.

XIII
Cosmotheoros

During the later part of the seventeenth century, Fontenelle,
historian of the Academic Royale des Sciences, attracted much
attention by writing descriptions of the earth as seen by hypo-
thetical inhabitants of Mars and Saturn. The little work, Cosmo-
theoros, written by Huygcns, and published posthumously in
1698, appears therefore to have been written in imitation of Fon-
teneJle and its chief interest is that it shows Huygens, at the
time of its composition, as notably Cartesian in outlook. It also
shows him, as Leibnitz had hoped, in a more human light than
most of his published work. The work was translated into
French by Dufour in 1702 and into English in 1722 when it
cnme out under the title The Celestial Worlds Discovr*d; it
is from this work that quotations have been taken.

Huygens began by remarking that scientific conjectures about
the planets should not be judged contrary to the scriptures nor
useless or impious. On the contrary, " besides the Nobleness and
Pleasure of the Studies, may not we be so bold as to say, they
are no small help to the Advancement of Wisdom and Morality,
so far arc they from being of no use at all? For here we may
count from this dull Earth, and viewing it from on high, con-
sider whether Nature has laid out all her Cost and Finery upon
this small Speck of Dirt." The English style of the translation
takes one far from the spirit of Huygens's Latin.

Nevertheless the work is throughout in an intimate vein,
being written in the form of a letter to Huygens's brother Con-
stantin. When Huygens outlined the Copernican theory he gave
a diagram " like what you have seen in my Clock at home ". Per-
haps he was referring to his machine for showing the planetary
motions. At all events, the reader recognizes much of Huygens's
early work in new dress the planetary magnitudes, the planets'
distances from the earth and his views on the earth itself as a
planet. "... we are so skilful nowadays," runs the translation,
" as to be able to tell how much more or less the Gravitation in
Jupiter or Saturn is than here . . . ". His general view was that
the planets are so like the earth in most essentials as to have

aoo

COSMOTHEOROS 2OI

inhabitants of some kind in all probability. Among these essen-
tials he reckoned the existence of water but perhaps with differ-
ent properties from our own. It must have a lower freezing point
on the cold planets. If some kind of human life exists, he sug-
gested, there must be other forms of life upon which the human
beings would be dependent. Man, he thought, on these other
planets, probably had the same vices and the same power of rea-
son. If his senses were not too different his studies were prob-
ably the same as ours. Huygens pictured the night sky as it would
appear to the inhabitant of Jupiter or Saturn. Letting his imagin-
ation go, he remarked: " What a wonderful and amazing Scheme
have we here of the magnificent Vastness of the Universe! So
many Suns, so many Earths, and every one of them stock 'd with
so many Herbs, Trees and Animals, and adorn 'd with so many
Seas and Mountains! And how must our Wonder and Admira-
tion be encreased when we consider the prodigious Distance and
Multitude of the Stars! " " I must be of the same Opinion with
all the greatest Philosophers of our Age," he added, "that the
Sun is of the same Nature with the fix'd Stars." He criticized
Kepler, therefore, for making the Sun superior to all other bodies
in the sky.

At the end he stated his modified vortex theory. " I am of
the Opinion," he wrote, " that every Sun is surrounded with a
Whirl-pool or Vortex of Matter in a very swift Motion; tho' not
in the least like Cartes'* either in their Bulk or manner of
Motion. For Cartes makes his so large, as every one of them to
touch all the others round them, in a flat Surface, just as you have
seen the Bladders that Boys blow up in Soap-suds do; and would
have the whole Vortex to move round the same way/' Descartes's
views, he asserted, needed to be corrected in the light of New-
ton's work, in particular to take account of the gravity of the
planets towards the Sun and how " from that Cause proceeds the
Ellipticity of the Orbs of the Planets, found out by Kepler ".
His own vortices, he explained, were composed of matter
which does not move all in the same way, "but after such a
manner as to have its Parts carry 'd different ways on all Sides.
And yet there is no fear of its being destroyed by such an
irregular Motion, because the ^Ether round it, which is at rest,
keeps the Parts of it from flying out."

Such were Huygens's last words on the solar system. To the
modern reader they seem curiously conflicting views. Huygens

202 HUYCENS'S SCIENTIFIC WORK

accepted all the mathematical part of Newton's work hut not his
interpretations. He was unable to accept a purely empirical view
of gravity and, as we have seen, considered his own experiments
on the existence of an ether were decisive. In his own way, Huy-
gens was only seeking to banish what he conceived to be occult
properties. " Le grand mrite de Descartes est avoir vu que le
probleme du monde est un probteme de mecanique," wrote
D'Alembert. This outlook was certainly the one which was con-
sistently adopted by Huygens. He did not see that his own work
was in far better accordance with the Newtonian system, that
it helped to expose and did not heal the wounds inflicted on Des-
cartes 's natural philosophy.

The Cosmotheoros reveals Huygens's religious outlook to a
far greater extent than his other published works. It does not
appear that he was ever a proclaimed rationalist, and indeed the
evidence rather shows, on the contrary, that he continued to
support Protestantism up to the end of his life. There were many
points, however, on which his beliefs appear to have been
unorthodox. While, for example, he considered that the consti-
tution of the world argued the existence of an intelligent power
behind phenomena, he felt that whatever divinity belonged to
man belonged to his rational mind; through this mind man can
apprehend the ways of the Creator, but equally this mind pre-
cluded an acceptance of the cruder superstitions. Huygens does
not seem to have accepted belief in the Devil, and perhaps he
rejected personal immortality. Yet his outlook clearly belongs
to that noble period of Protestant thought which found nothing
alien in the new and enlarged horizons revealed by Science. As
an admirer of the writings of Cicero, Huygens refers to the glory
to be found in Nature as in some way a warranty that man is not
without his significance in the scheme of things. Stoic philosophy
found a response within the breast of this seventeenth century
scientist, and one is reminded that from the beginning of the
Christian era there had been an infiltration from this philosophy
of the grand belief in the ultimate rationality of the world. This
belief Huygens certainly supported; it was, one might say, the
essence of his religious outlook.

XIV
The Place of Huygens in the History of Science

Descartes and Galileo were the brightest stars in the scientific
firmament at Huygens's birth and they influenced him one way
or another all his life. In Huygens's early days the world of
science may be said to have been divided into those who followed
the empiricism of Galileo and those who, with Descartes,
ultimately distrusted it. This division leaves on one side, however,
the Aristotelians who, mainly for religious reasons, could not
accept Descartes's ingenious reconciliation of Copernicanism
with dogma. The universities which, on the whole, sheltered the
Aristotelians were in consequence left behind by the faster-mov-
ing currents of contemporary thought. In time, it is true, they
became affected by Descartes's thought and this acted as an
introduction for more scientific ideas. Newton's views were first
introduced at Cambridge as a sort of commentary on Descartes;
at Oxford, the Sjivilian professorship of Astronomy almost alone
was renowned for progressive ideas. The scientific societies of
the early part of the seventeenth century were unacademic,
amateur, spontaneous offshoots of the artistic renaissance, owing
their existence partly to the rediscovery of Greek writings of
analytical character, partly to the conflict of ideas in Astronomy
and the development of Mathematics, and partly to the writings
of Francis Bacon and of Descartes. The sources of the scientific
renaissance include other tributary streams but when all is said
the explanations seem incomplete and hardly concern us here.
The important fact is that Huygens was born in the period when
these scientific societies were in their infancy. As a young man,
he must have heard of the work of the Florentine Accademia del
Cimento and of the interest kindled in Paris through the reports
of the well-known traveller Pieresc, who visited Florence.
Huygens early studied the works of Galileo and through
Mersenne obtained his introduction to a small world which was
preoccupied with new problems and becoming more and more
confident that it possessed a new technique of discovery.

For the attack on Aristotle's science had been long prepared.
Roger Bacon, Da Vinci, Benedetti and Stevinus preceded Galileo

203

204 HUYGENS'S SCIENTIFIC WORK

and began the initial liberation of Mechanics. The first criti-
cisms, however, dealt with certain of Aristotle's postulates rather
than with his entire method and Galileo was the first to replace
the substance and attributes of scholastic description by what
we must term scientific data. The whole movement of the seven-
teenth century empiricists, then, took this direction. Instead of
substance, essence, matter, form and other categories adapted to
Aristotle's logic, an analysis was developed using those of
space, time, mass, force and the like. The categories of
thought were transformed. All this Huygens himself realized
remarkably clearly. He perceived that even the sixteenth century
writers had retained many of the occult properties of the Aris-
totelians. Gilbert, Telesius and Campanella, he noted, had not
enough inventiveness or mathematics. Even Gassendi was not
much better. Bacon had seen the insufficiences of Aristotle and
had in addition pointed out good methods for building a better
system, but, he wrote, " he did nothing to advance mathematics
and lacked penetration for physical matters, not having been
able to conceive the motion of the Earth, at which he mocked *'.
" Galileo, on the other hand, had the mind and all the knowledge
of mathematics he needed to make progress in Physics and it
must be admitted that he had been the first to make fine dis-
coveries touching the nature of movement, although he left very
considerable parts of it to be done. He had not sufficient boldness
nor presumption to wish to undertake the explanation of all
natural causes, nor the vanity to wish to be chef de secte. He was
modest and loved truth too much; he believed besides that he had
acquired enough fame and that, through his new discoveries, it
would last for ever."

Another feature of contemporary scientific thought was the
assumption of a mathematical simplicity in the relations trace-
able between data. This assumption is found in the work of
Galileo and also in that of Kepler, Copernicus and, to some ex-
tent, perhaps, even in that of Da Vinci. Kepler expressed the
idea in two aphorisms : " Natura simplicitatem amat " and
" Natura semper quod potest per faciliora non agit per ambages
difficiles." Burtt has pointed out 1 that the decline of Aristotel-
ianism in the sixteenth and seventeenth centuries coincided with
a rise of neo-Platonism in which there was a strong Pythagorean
element. For Kepler, in the extreme instance, the mathematical
1 The Metaphysical Foundations of Modern Science (1932).

HUYGENS'S PLACE IN SCIENCE 205

harmony discoverable in the facts of Nature and even in
the celestial regions was the reason why things are as they are.
Nevertheless, this neo-Platonic mysticism was combined, in Kep-
ler, with a reverence for exactitude in the mathematical
formulation. Later this view was greatly modified; the
existence of mathematical regularity was felt to indicate a
mechanical explanation. The difficulty lay in combining the
mathematical laws with a mechanism which would sacrifice none
of their exactitude. Descartes's mechanism failed here. His cos-
mology was in fact founded on an antithesis which was truly
scholastic, since he tried to reconcile Copernicanism with an un-
moved earth an earth at rest in its heaven. Huygens became
convinced that Descartes in this and other matters had in fact
repeated the errors of Scholasticism, for he hoped to found a de-
monstrative and deductive system. Descartes, he noted, " who
appears to me to have been very jealous of the renown of Galileo,
had this great desire to pass for the author of a new philosophy.
Which was clear from his efforts and his hopes to have it taught
in the academies in place of that of Aristotle . . . ". Descartes's
ideas, he admitted in another passage, were presented with all the
force of verified conclusions; the novelty of the shapes of his
ultimate particles of matter and the beauty of his vortices all
exerted a compelling influence. " It seemed that when I read this
book, the Principia, the first time that everything in the world
became clearer, and I was sure that when I found some difficulty,
that it was my fault that I did not understand his thought. I was
then only fifteen or sixteen years old. But having since dis-
covered from time to time things clearly false and others very
improbable, I came back strongly to the preoccupation I was in,
and at the present time I find scarcely anything I can accept as
true in all the physics, metaphysics and meteors/'

Huygens, in fact, returned to the outlook of Galileo. He had
learnt that quantitative study of data and not scholastic logic fur-
nished the technique of discovery. Descartes's powers as a mathe-
matician compelled Huygens's admiration and he found it all the
more unforgivable that exact agreement with the facts should
not be uppermost in Descartes's work. The difference between
Descartes and Huygens did not lie in their conception of the phy-
sical processes so much as in the regard paid to accurate defini-
tion of physical conditions. More than geometry was needed,
Huygens perceived, to deal with this. There were more properties

2o6 HUYGENS'S SCIENTIFIC WORK

than mere extension, which was quite inadequate to account for
the results of collision between elastic bodies and for accelerated
motion.

These considerations are, however, as yet inadequate as an
account of the revolution in Mechanics which we have been led
to discuss. For the change in mental categories and the expres-
sion of laws in corresponding mathematical form was combined
with a profound change in the method of explanation involved
in all scientific work and especially in Mechanics. In broad
terms, the revolution against Aristotelianism was the rejection
of Aristotle's final cause in favour of a scientific mode of his
efficient cause; men asked not what purpose but what process lay
behind phenomena. How radical this change was we can realize
only if we remember that for Aristotle, with his biological out-
look, all events were part of a natural process of fulfilment, of
realization of what was innate. Things converged towards an
appointed and necessary end; it was consequently more interest-
ing to enquire what that end is, rather than what mechanism
underlies events. In Dynamics all motion was motion to an
appointed place; all nature bore the character of an innate im-
pulse to movement.

Movement and change set greater problems for the philoso-
phy of Plato. Whereas movement was regarded as continuous by
Aristotle, to Plato all change appeared as a succession of Forms
and consequently movement must be essentially discontinuous.
It may well be, therefore, that for an Aristotelian of the sixteenth
century movement held less interest and less to be studied than
it did for those who were influenced by the revival of neo-Platonic
ideas.

There was, however, a second reason for the revolt against
Aristotle, whose authority so dominated the scholastics. Quite
apart from the difference in the nature of cause itself, there was
the objection that no knowledge of the world as it is could be
derived by deductions from a priori metaphysical principles.
Aristotle's system was out of touch with brute facts, his method
unsuited to their investigation. Aristotle, in fact, did not see that
induction demanded a correlating idea; that it could not
be reduced to some kind of syllogism, and consequently that know-
ledge of the world in the last resort cannot be made part of
logic. Bacon, though not a scientist, was clear on this " unfruit-
fulness" of Aristotle and expressed the new attitude to facts in his

HUYGENS S PLACE IN SCIENCE 2O?

famous aphorism : " Nature to be commanded must be obeyed".
It became clear from this time onwards that much remained to be
discovered in respect of even the simplest events.

Nevertheless, it is clear that what we now hold to be the
typical method of the physical sciences appeared then to be a
much more individual question. Galileo had given an excellent
example but Huygens carried the inter-relation of mathematics
and experiment a long way further. All his work illustrates this
quality, and certain continental writers have even argued that
Huygens's conception of scientific method was in some respects
superior to that of Newton. Certainly in regard to the position
of hypotheses in scientific work a case may be made out for
Huygens's superiority. "... the main Business of natural Philo-
sophy is to argue from Phaenomena without feigning Hypo-
theses, and to deduce Causes from Effects, till we come
to the very first Cause, which certainly is not mechanical," wrote
Newton in the twenty-eighth query of his Optics. Against this
we might place Huygens's remarks on the essential place of
hypothesis in scientific work which come at the beginning of his
Traite de la Lumiere.

It was in the use of abstractions that Newton made the
greatest contribution. He had the strongest objection to hypo-
thetical entities because he wished to concentrate on mathematical
relations. The space and time of Newton's system were not
identical with the space and time of ordinary experience; they
were abstractions. Huygens was unable to understand Newton's
more positivist attitude here. The realist nature of his own con-
ceptions makes him, in fact, a convenient starting point from
which to trace the second main stream of scientific thought.
Huygens might have accepted the term " correlate " as equivalent
to hypothesis but, in the manner of all atomists, his concepts
borrowed a garb from perceptual phenomena. For him the atom
was a potential phenomenon, as it still is for some modern men
of science. He did not see as clearly as Newton that for Science
what is needed is a number of principles of quantitative correla-
tion. This, surely, is the true end of scientific induction. In
practice, in the scientific interpretation of phenomena not capable
of treatment by classical mechanics, much use is always made of
entities which must possess the qualities of hypotheses. This
method Huygens, if not Newton, would have supported. We
may summarize this discussion by saying that the Huygensian

2O8 HUYGENS'S SCIENTIFIC WORK

method confers greater freedom than the Newtonian and, in
regard to the broad conception of the place of hypotheses,
Huygens was a more profound methodologist than Newton.

Huygens's work was very influential in its day and it is cer-
tain that Newton was at least stimulated by him. An interest-
ing example is Huygens's account of the ether setting aside,
for the moment, the distinction of the subtle matter. Newton
did not accept this sort of explanation of gravity but equally
he recognized the difficulties of the so-called "action at a
distance ". He believed with justification that his Principia dis-
posed completely of the Cartesian vortices of subtle matter and
he returned to this subject in his Optics. "... against fill-
ing the heavens with fluid mediums, unless they be exceeding
rare, a great objection arises from the regular and very lasting
motions of the planets and comets in all manner of courses
through the heavens. For thence it is manifest, that the heavens
are void of all sensible resistance and by consequence of all sen-
sible matter/' But Newton never denied that an ether might exist
and that it might be conceived as a medium susceptible of trans-
mitting vibrations. Burtt 1 even writes: " Halving taken over the
notion from the current of the times, and feeling it to be thus
well grounded, it was easy for Newton to extend its use to other
phenomena which involved action at a distance and which others
were accounting for in the same fashion, such as gravity, mag-
netism, electric attraction, and the like." Newton was, however,
not at all consistent on the subject of the ether and it is clear
that his conception of it was different from that of Hooke. There
is some probability that it was Huygens's work on the subject
and particularly the experiments using the vacuum pump which
influenced him most. Accounts of Huygens's experiments on
the non-descent of columns of water and mercury were common
in England, notably in the Philosophical Transactions, and the
observations were also made by Boyle, Brouncker and others.
Most of Newton's comments on the ether are to be found in his
Optics, in which they first appeared in the Latin edition of 1706.

The only acknowledged debt of Newton to Huygens was the
statement of the theorems of centrifugal force in the Horologium
Oscillatorium. Newton, as L. T. More has remarked, " must have
seen that Huygens's law of centrifugal force was easily deduciblc
from his own calculation on the attraction of the moon, and that
* E. Burtt. op. cit. p. 165.

HUYGENS'S PLACE IN SCIENCE 209

by neglecting to follow up his work, Huygens had preceded him."
This is suggested by Newton's statement : " What Mr. Huygens
has published since about centrifugal forces I suppose he had
before me." The view which is now most widely held in regard
to the difficult problem of the order of Newton's ideas is that
he solved the problem of centrifugal force independently of Huy-
gens but neglected to publish anything until he had proved the
important theorem concerning the gravitational field due to a
large solid sphere. Newton was uninfluenced by Huygcns's very
important development of the conceptions ot energy and work
done in mechanical systems. An account of the growth of liuy-
gens's ideas has been given in the sections on impact and on the
centre of oscillation. 1 Unfortunately, the only direct influence
of Huygens's ideas is to be found in the work of Leibriirzians,
who made the conservation of vis viva a cosmic principle instead
of treating it, as Huygens did, as only half the true law of the
conservation of mechanical energy. Huygens did not go so far
as Newton towards complete scientific positivism but he agreed
that mathematical law is in itself the most important end of
scientific work. For Newton the essential aim of science, in his
own words, was to replace " occult properties supposed to result
from specific Forms of Things " by " general laws of Nature ".
The wisdom of this limited aim has been fully demonstrated.

If Huygens saw the direction in which Leibnitz's ideas were to
lead philosophy it is certain that he could not have sympathized
with them. With his increasing recognition of Descartes's errors
in purely physical matters, Huygens, as we know, paid less atten-
tion to his philosophical system and so felt unperturbed by the
dilemma of his dualism. If this had not been so, Huygens might
have inclined to a materialism of the kind elaborated by his con-
temporary, Hobbes, but it is inconceivable that he would have
tolerated many of the latter's crudities. Huygens was something
of a materialist and he rejected the orthodox religious doctrines:
nevertheless, if he had written philosophy it would have been
without the harshness of Hobbes's determinist schemes.

Yet the bifurcation of the world into physical and mental
spheres was as much the work of men like Huygens as it was of
Descartes, who first presented the bifurcation in its most uncom-
promising form. Kepler took the first step of distinguishing pri-
mary from secondary qualities. For him only those qualities

1 Nature, 1943, p. 519.
O

2IO HUYGENS S SCIENTIFIC WORK

which could be measured were primary and this attitude was
found in Galileo, who defined the two classes more clearly and
made the corresponding distinction between true knowledge (of
primary qualities) and mere opinion. Colour, smell, taste and
sound were for him matters of opinion subjective impressions
resulting from the operations of atoms or vibrations on the sense
organs. This set the stage for the Cartesian dualism. It is not
perhaps properly recognized how far Huygens contributed to the
adoption of Galileo's attitude. But we have only to recall that
Huygens's whole work was the reduction of more phenomena to
quantitative treatment their transformation from matters of
opinion to those of knowledge. Scientific time measure for short
intervals began with Huygens; he related the standard of length
with that of time, he provided a geometrical treatment of reflec-
tion and refraction of light, he greatly extended Mechanics and
elucidated the true nature of more celestial phenomena in accord-
ance with Copernicanism all this is a direct continuation of
lines of thought to be found in Galileo. Through him, in fact,
the main stream of scientific thought may be said to have been
diverted from following Descartes and instead directed into the
channel which Newton's work deepened to a river. But Huygens
would not have had us forget the imaginative stimulus of Des-
cartes 's writings. As late as 1691 he remarked: "We owe
much to Descartes because he revealed new paths in the study
of Physics and started the idea that everything must be reduced
to mechanical laws/'

Huygens came nearest to explaining his own method in
scientific study in a letter to Tschirnhaus in 1687. ^ n thc problems
of Physics, he wrote, very great difficulties are felt at first and
these cannot be overcome " except by starting from experiments
. . . and then by conceiving certain hypotheses . . . But even
so very much hard work remains to be done and one needs not
only great perspicacity but often a degree of good fortune." One
is reminded of Whitehead's remark ". . . it is the establishment
of the procedure of taking the consequences seriously which
marks the real discovery of a theory ". At times Huygens
followed his own procedure of taking the consequences seriously
up to the point at which his elegant geometrical constructions
were threatened. For him, as for Kepler, there seems to have
been a disposition to believe that mathematical elegance is in
some way an index to reality. This feeling for form may be

HUYGENS'S PLACE IN SCIENCE 211

found in many men of science but in few has it been so marked
as in Huygens. This sense for elegant theoretical construction,
roused by the symmetry and order which can be found in
Nature, is well illustrated by the research on Iceland spar in
which one has, as it were, the essence of Huygens's thought. Yet
it was the weakness which predisposed him to believe in the
rotation of the matiere subtile around the earth. He agreed with
Leibnitz's comparison of Galileo and Descartes : " Galileo excels
in the art of reducing mechanics to science; Descartes is admir-
able at explaining by beautiful guesses the causes for the effects
of nature." He agreed, too, with Leibnitz's remark that
Descartes's work was " un beau roman de Physique " ; one feels
that it was with regret. And yet, without Huygens's careful
studies how could Leibnitz have coined that phrase?

PERSONS MENTIONED

Many of the persons mentioned in the first part of this book will
be unknown to the general reader, and some of them are but little
known to the historian. The following notes provide some of the chief
facts known about them.

Auzout, Adrien (d. 1691). French mathematician and astronomer.
Auzout is chiefly remembered for his invention of a micrometer con-
sisting of movable hairs, mounted parallel in the field of view of the
astronomical telescope. With Picard (below) he was a pioneer in the
application of the telescope to graduated scales, thus replacing the use
of open sights upon which Tycho Brahe had had to rely. He worked
in Paris with Huygens and helped in the development of the
" aerial " telescope.

Barrow, Isaac (1630-77). This noted English mathematician was
the first holder of the Lucasian professorship at Cambridge; on his
arrival he became Newton's tutor and he undoubtedly influenced
Newton through his own interests in geometrical optics and mathe-
matics. Previously he had had a somewhat eventful life, his political
and religious views as a young man in his twenties making it desirable
that he should travel on the Continent. He became F.R.S. in 1662,
the year before he went to Cambridge. In 1669 he resigned the
Lucasian chair in Newton's favour, desiring among other things to
devote himself to theology.

Boulliau, Ismael (160594). French astronomer. Before working in
Paris, where he met Huygens, he spent some time travelling in Italy,
Holland and Poland, becoming acquainted with the leading men of
science. He was a friend of Leopold de Medici and corresponded
with him on scientific matters of the time.

Campanella, Tomasso (1568-1639). He does not come into the
period covered by this book, but his writings and opinions were well-
known to Huygens and he is to be remembered as one of the earliest
champions of the experimental method. His most famous work is his
Defence of Galileo (1622), in which he quoted the great religious
thinkers from Augustin to Aquinas to show how unjustified the
persecution of Galileo was.

213

214 THE LIFE OF CHRISTIAN HUYCENS

Carcavy, Pierre de (d. 1684). Did important administrative work in
Paris during the period under review. Colbert made Carcavy librarian
to the King (1663) and it was during Carcavy's period of office that
the royal library was moved to new rooms which became also the
meeting place of the Academic Royale des Sciences. Carcavy's
mathematical knowledge was good and he took part in 1645 * n tne
dispute over the quadrature of the circle, claiming to demonstrate
that the quadrature was impossible. He was a friend of Pascal,
Descartes, Roberval and Fermat.

Cassini, G. Domenico (1652-1712). Born at Perinaldo near Nice,
and educated by the Jesuits at Genoa, Cassini at twenty-five became
professor of astronomy at the famous university of Bologna. Here he
became distinguished and his fame spread to Paris when he published
ephemerides for the satellites of Jupiter. Through the efforts of Picard
he was invited to Paris and after 1671 he was virtually the director of
the Paris observatory. He collaborated with Richer in 1672 on the
observations of the parallax of Mars. With Picard and Auzout he
did a great deal towards founding the great tradition of the Paris
observatory. His most famous observation was the discovery of the
satellites of Saturn and the division in its ring now known by his name.

Fermat, Pierre de (1601-65). One of the greatest of French mathe-
maticians. Fermat anticipated Descartes's invention of analytical
geometry in nearly all respects, and he contributed important work on
the treatment of maxima and minima. With Pascal he laid the
foundations of the mathematical study of probability (Huygens also
wrote on this subject), and he discovered the important " least-time
principle " in optics. He adversely criticized some of Descartes's
optical theory.

Frenicle de Bessy (1605? 1675). Frenicle held an official position in
Paris, where he acquired a great reputation as a demon calculator. He
could solve the most complex numerical problems in arithmetic and
so quickly as to astonish even Fermat, Roberval and Wallis. His
method of working was kept a secret and was examined after his
death. His " method of exclusions ", which he used so much, is now
only an object of curiosity and he contributed nothing of permanent
value.

Hevelius(or,]ohann Hevel) (1611-87). Noted astronomer of Danzig.
Although he made first-class observations of the planets and the moon
employing telescopes, he preferred to make all his measurements
with open sights. This led to a dispute with the English astronomers
and he was visited in Danzig by Halley, who observed with him and
employed telescopic sights. Neither astronomer converted the other.

PERSONS MENTIONED 215

Mariotte, Edm& (1620-84). A Roman Catholic abbe with a weak-
ness for experimentation. He joined the Academic Royale des
Sciences in the year of its foundation. His most valuable work was
on impact; he independently enunciated the law known as Boyle's
law fcf. his Discours de la Nature de VAir (1676)].

Papin, Denis (16471712). Had a rather precarious existence and
died in obscurity in London. For a time he assisted Huygens, notably
with his experiments on the air pump and on the expansion of steam
as a source of power. Later he came to London to work with Boyle
and in 1684 he became temporary curator to the Royal Society. In
1687 he was appointed professor of mathematics at Marburg. From
there he went to Cassel (1696) and then on to London (1707).

Petit, Pierre (1594-1677). An able civil servant who was given his
chance by Richelieu. He was quite a distinguished mathematician
and he had quite a lot of contact with Huygens during the latter's
early visits to Paris. His daughter was attractive. A friend of Pascal
also, he helped him in his experiments on the vacuum.

Picard, Jean (1620-82). One of the most famous astronomers ot
the seventeenth century. He first of all observed with Gassendi, whom
he replaced as professor of astronomy at the College de France. In
1666 he became an abbe* but without any interruption of his scientific
output. In 1669 he read to the Academic an important memoir on the
new methods in astronomy resulting from work by Huygens and
himself. He went to Uraniborg, Tycho Brahe's famous observatory
(of which nothing then remained), in order to fix more accurately its
latitude and longitude. He met Roemer on this journey and got him
to come back to Paris with him. Picard's Mesure de la Terre was an
important work and provided Newton with the information he needed
in his calculations on the earth's gravity.

Roberval, Giles Persone de (1602-75). A French mathematician
who rose to eminence from obscure origins. He came to Paris in 1627
and stayed with Mersenne. In 1631 he was appointed to a chair of
philosophy and then to the professorship of mathematics at the
College de France. He was one of the original members of the
Academic Royale des Sciences. After 1638 he had strong differences
with Descartes and became hostile to Descartes's philosophy. He was
a man of decided originality of mind and his mathematical work is
of importance.

2l6 THE LIFE OF CHRISTIAN HUYGENS

Roemer, Glaus (16441710). Noted Danish astronomer. Roemer
worked at first at the Round Tower of Copenhagen, which had been
built by Christian IV for one of Tycho Brahe's assistants. He is
chiefly remembered for bringing the transit telescope into general
use. His calculation of the velocity of light is also famous. After his
death his instruments and records were destroyed in the great fire of
Copenhagen (1728), but it was found possible to reconstruct much
of what was lost.

Sorbiere, Samuel de (1615-70). French writer. Sorbiere was in-
tended at first for the Protestant ministry and was brought up by an
uncle after the death of his parents. He studied medicine in Paris
and practised in Holland up to about 1650. In 1653 ^ e became a
Catholic, and in 1655 he went to Rome and was received by the Pope.
Lacking any more solid rewards, Sorbiere sought patronage in Paris
and became loosely attached to the circle of scientific amateurs. He
was not thought to have much originality or learning by his
contemporaries.

Wallis,John (1616-1703). Famous English mathematician. Wallis
was ordained in the English Church after completing his time at
Emmanuel College, Cambridge, and he then served as chaplain to a
noble family. His fortune was made by his discovery of his ability to
decipher codes for the parliamentary party during the civil war.
In 1649 Cromwell appointed him Savilian professor of Geometry at
Oxford. He became a friend of Newton and his Arithmetica
Inftnitorum led the latter direct to his discovery of the binomial
theorem. He became associated with Boyle and helped to found the
Royal Society.

A SHORT BIBLIOGRAPHY

IIUYGENS'S LlFE AND TlMES.

There is a wealth of information in Huygens's correspondence
contained in Volumes i to 10 inclusive of the Oeuvres Completes de
Christiaan Huygens, published by the Socie'te Hollandaise des
Sciences. The notes given in Part I of this book represent a very small
selection of this mass of information.

Also very useful for the student is Le Sejour de Christian
Huygens a Paris by H. L. Brugmans (1935).

SCIENCE IN THE SEVENTEENTH CENTURY.

WORKS OF A GENERAL CHARACTER.

A History of Science, Technology and Philosophy (i6th and I7th
Centuries) by A. Wolf. (Allen and Unwin, 1935).

The Metaphysical Foundations of Modern Science by E. Burtt.
(Kegan Paul, 1932).

Ceschichte der Physik by J. C. Poggendorff, (1879).

The Role of Scientific Societies in the Seventeenth Century by
M. Ornstein (Chicago, 1928).

Scientific Organisations in Seventeenth Century France by H.
Brown (Baltimore, 1934).

WORKS OF MORE SPECIALIZED INTEREST.

Le Developpement de la Physique Carttsienne, 1646-1712 by P.
Mouy (Paris, 1934).

Isaac Newton by L. T. More (Scribners, 1934).

Les Origines de la Statique by P. Duhem (1905).

La Theorie Physique by P. Duhem (1906).

The Science of Mechanics by E. Mach (English trans, by T. J.
McCormack, 1919).

The Mathematical Principles of Natural Philosophy by I.
Newton (English trans, by F. Cajori, 1934).

Discourses on Two Sciences by G. Galilei (English trans, by
Crew and De Salvio, 1914).

A History of the Theories of /Ether and Electricity by E. T.
Whittaker (1910).

Optics by L Newton (Bell, 1931).

Matter and Gravity in Newton's Physical Philosophy by A. J.
Snow (1926).

2l8 THE LIFE OF CHRISTIAN HUYGENS

FOR HUYGENS'S WORK IN MECHANICS, see :

Die. Pendeluhr by Christian Huygens (German trans, by
Heckscher and Oettingen, 1913), Ostwald's Klassiker der Exakten
Wissenschaften No. 192.

Oeuvres Completes, vols. 16, 17, 18, 19.

The Evolution of Clockwork by J. Drummond Robertson (1931).

FOR HUYCENS'S WORK IN OPTICS, see:

Treatise on Light by Christian Huygens (English trans, by S.
Thompson, 1912).

Oeuvres Completes, vols. 13, 19.

The Principles of Physical Optics by E. Mach (English trans, by
Anderson and Young, 1926).

FOR HUYGENS'S WORK IN ASTRONOMY, see :

Oeuvres Completes, vols. 15, 21.

Histoire de lAstronomie Moderne by A. Delambre (1821).

This bibliography is limited to comparatively modern publica-
tions. The references to the Oeuvres Completes de Christian H
Huygens are not intended to be complete, but they indicate the
volumes in which the greater part of a particular subject is treated.
There is, in addition, the work on Mathematics to he found in vols.
u, 12. 14, 20

INDEX

ACADEMIE ROYALE DES SCIENCES, 41

54, 58, 78

Accademie del Cimento, 36, 203
Anagrams for publication of discov

cries, 32

Aristotelianism, 52, 97
Atomic theory, 28, 94, 101
Auzout. Adrien, 41, 57, 59, 213

BACON, FRANCIS, 46, 62

Barrow, Isaac, 65, 213

Benedetti, 21, 99

Boulliau, Ismael, 27, 32, 210,

Boyle, Robert, 48, 57, 9:?

Brahe, Tycbo, 35, 127

Breda, College of, 20

Bruce, Alexander, Earl of Kincardine,

CAMPANELLA, THOMASSO, 18, 204, 213
Campani, Guiscppe, 44, 56
Cartesianism, 15, 20, 24, 60, 68, 8,

86, 109
Cassini, Domenico, 40, 43, 56, 63, 77

102, 214

Centre of gravity, 112, i*5> 14^
Centre of oscillation, 150
Centre of percussion, 22, 150
Centrifugal force, 64, 117, 161
Chapelain, Jean, 27
Chemistry, 93
Chromatic aberration, 173
Circular motion, 63, 85
Clock, pendulum, 34, 127; in astron

omy, 39, 133; in determining longi

tudes. 40, 54, 83, 89, 135; spring

regulated, 55, 70
Colbert, 6, 41/46, 83
Colour, 57, 71
Comets, 89

Compound pendulum, 99, 150
Conical pendulum, 69, 120
Conrart, 27

Conservation of energy, 114, 154
Conservation of momentum, no
Copernicanism, 34, 44, 99, 193
Cosmotheoros, 89, 200
Coster, Samuel, 37
Cvcloidal pendulum, 37, 39, 136

De Motu ex Pefcusu'one, i 1 1

De Vi Ccntrifuga, 117

Descartes, Ren6, 14, 18, ioj, 110. 176

Diffraction, 174

Divinis, Eustachio de, 34, 44

Double refraction, 74, 183

EARTH, FORM OF, 122

Eclipses, of sun, 62

Energy, conservation of, 114, 154

Ether, 162, 177, 180, 208

Evolutes of curves, 146

Eyepiece, Huygens's, 52, 171

FABRI, HONORED 34, 155
Fermat. Pierre de, 81, 182. 214
Fermat's principle, 58, 182

GALILEO, 22, 30, 98, 136, 150
Gassendi, Pierre, 27, 105
Gilbert, William, 101, 161
Gravity, 33, 63, 85, 90, 117, 161
Gresham College, 41, 44, 54

HEVELIUS, JOHANN, 30, 62, 214

Hobbes, Thomas, 46, 209

Holmes, Captain, 51

Hooke, Robert, 55,' 70, 73, 144; Micto-
graphia, 57, 65, 176; wave theory,
77, 176

Horo/ogmm, 35, 128

Horologium Oscillatorium, 40, 61, 67,
131, 138

Huygens, Christian, character, 52, 60;
education, 7, 19; health, 7, 60, 65,
75 94 as mathematician, 25, 68,
81, 91; conception of scientific-
method, 73, 178, 207, 210; mechan-
istic outlook, 9, 79, 162; and New-
ton, 15, 71, 72, 84, 209; and Leib-
nitz, 86, 88; and Hooke, 55, 70, 73:
and Spinoza, 9, 62; and Bayle, 87;
and Huet, 87; in Paris, 7, 29, 43,
58, 67, 77; in London, 45, 50, 53,
84; religious outlook, 7, 94, 202; on
universal gravity, 86, 89; atomic
theory, 28, 94; and reflecting tele-
scope, 71, 93; eyepiece, 52, 171;
work in optics, 74, 81, 167

Huygens, Constantin (the elder), 16:
(the younger). 19, 9*

319

210

THE LIFE OF CHRISTIAN HUYGENS

ICELAND SPAR, 74, 177, 184

Impact, theory of, 64, 109
Intercepted pendulum experiment,
121

JORDANUS NEMORARIUS, 125
Journal des Savants, 55
Jupiter, 36

KEPLER, 98, 165

LEAST-I IMF. PRINCIPLE, 81, 182
Leibnitz 69, 80, 86, 88, 90
L ens-grindiug, 51
Leopold de Medicis, 33, 35
Lcyden University, 19
Light, propagation of, 76, 176
Longitude determination, 35, 89, 13*
Louis XIV, 47, 61

MARIOTTE, EDM, 64, 164, 215

Mars, 103

Mersennc, Marin, 21, 25, 100, 138

Micrometer, 57, 197

Microscope, 65, 82, 174

Moment of inertia, 151

Momentum, 64

Montmor Society, 41, 43, 53

NEWTON, and centrifugal force, 64,
09; the Principta, 84, 89; and the
ether, 85, 208; and the Horologium
Oscillatorium, 68; and Huygens, 15,
71, 72, 84; and his reflecting teles
cope, 70; and colours, 71

OBSERVATORY AT PARIS, 63
Oldenburg, Henry, 45, 53, 65, 71, 78

PAPIN, DFNLS, 67, 74, 190, 215
Pascal, 23, 38, 43, 147
Philosophical Transactions, w
Picard, Jean, 59, 67, 103, 215

Planetary machine, 83
Polarized light, 189
Potver, Thomas, 8

RtFRAMIVE INDEX, 167, l8l

Richer, 103

Roberval, Giles Persone de, 25, 32,

164, 215

Roemer, Olaus, 39, 77, 83, 216
Renault, Jacques, 95, 116
Royal Society, 41, 53, 66

SATURN, 26, 30, 44, 52, 193

Schootcn, Frans van, 19, 25

Simple hiirmonic motion, 99, 143

Sorbierc, Samuel dc, 28, 41, 53, 216

Space, absolute, 122

Spherical aberration, 171

Spino/a, 9, 62

Statics, 124

Stevin, Simon, 19, 21

Subtle matter, 49, 163, 178

Sy sterna Saturniurn, 33, 195

TAUFOCHRONE, 136

Telescopes, 26, 30, 45, 56, 193; aerial,

51, 56

ThcVenot, Melchisc'dech, 41, 54
Thuiet, 38

TraitS dc la Lumiere, 49, 76, 81, 178
Tschirnhaus, Walter von, 75

VACUUM, 23, 163

Viviani, 23

Voorbutg, 55

Vortex theory, 24, 86, 90, 109, 163, 201

WALLIS, JOHN, 45, 115, 126, 216
Wave theory of light, 76, 77, 176
Weight and mass, 119
Wren, Christopher, 52